The Weight-Enumerators of Domino Tilings of Holey Rectangles with thin Boundaries up to width, 3 By Shalosh B. Ekhad [ShaloshBEkhad@gmail.com ] ------------------------------------------------------------------- Theorem Number, 1, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 1, d[1] = 1, d[2] = 1 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- 2 2 \ n -1 - h - 2 h t ) A(n, h) t = ------------------- / (h t - 1) (h t + 1) ----- n = 0 and in Maple input format: (-1-h^2-2*h^2*t)/(h*t-1)/(h*t+1) We now present a statistical analysis Let, b, be the algebraic number, 1 whose minimal polynomial is, b - 1 and whose floating-point appx. , to, 10, digits is, 1. We have the following proven facts The total number of tilings of the region is asymptotic to: n 2 (1/b) that in floating-point is: n 2 1. and in Maple input format: 2*1.^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 1 + n and in Maple-input format: 1+n and in floating point: 1 + n The asymptotic expression for the variance, as a function of n is: 1/2 and in Maple-input format: 1/2 and in floating point: 1/2 The even alpha coefficients until the, 8, -th , are: [1, 2, 4, 8] and in Maple-input format: [1, 2, 4, 8] and in floating-point it is: [1, 2, 4, 8] The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) This ends this theorem. ------------------------------------------------------------------- Theorem Number, 2, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 1, d[1] = 1, d[2] = 2 Then infinity ----- 2 2 2 2 7 2 2 \ n -h (2 + h ) + h (2 h + 1) (1 + h ) t - h (1 + h ) t ) A(n, h) t = ------------------------------------------------------- / 3 6 2 3 ----- (h t - 1) (h t - 2 h t - h t + 1) n = 0 and in Maple input format: (-h*(2+h^2)+h^2*(2*h^2+1)*(1+h^2)*t-h^7*(1+h^2)*t^2)/(h^3*t-1)/(h^6*t^2-2*h^3*t -h*t+1) We now present a statistical analysis 1/2 5 Let, b, be the algebraic number, 3/2 - ---- 2 2 whose minimal polynomial is, b - 3 b + 1 and whose floating-point appx. , to, 10, digits is, 0.381966012 We have the following proven facts The total number of tilings of the region is asymptotic to: / 4 b\ n |11/5 - ---| (1/b) \ 5 / that in floating-point is: n 1.894427190 2.618033984 and in Maple input format: 1.894427190*2.618033984^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 6 b / 4 b\ 7/5 + --- + |9/5 + ---| n 5 \ 5 / and in Maple-input format: 7/5+6/5*b+(9/5+4/5*b)*n and in floating point: 1.858359214 + 2.105572810 n The asymptotic expression for the variance, as a function of n is: 28 24 b /24 16 b\ -- - ---- + |-- - ----| n 25 25 \25 25 / and in Maple-input format: 28/25-24/25*b+(24/25-16/25*b)*n and in floating point: 0.7533126285 + 0.7155417523 n The even alpha coefficients until the, 8, -th , are: 23 3 b 39 b -- - --- 19/2 - ---- - 3/2 + b 20 2 13 b - 39/2 2 [1, 3 + --------- + --------, 15 + ----------- + -----------, n 2 n 2 n n -231 + 154 b - 119/4 - 231 b 105 + ------------ + ---------------] n 2 n and in Maple-input format: [1, 3+(-3/2+b)/n+(23/20-3/2*b)/n^2, 15+(13*b-39/2)/n+(19/2-39/2*b)/n^2, 105+(-\ 231+154*b)/n+(-119/4-231*b)/n^2] and in floating-point it is: 1.118033988 0.5770509820 14.53444184 2.051662766 [1, 3 - ----------- + ------------, 15 - ----------- + -----------, n 2 n 2 n n 172.1772342 117.9841488 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 3 b - 3/4 + --- 3/10 - b/5 10 30 - 20 b -110 + 30 b [---------- + -----------, --------- + -----------, n 2 n 2 n n 6615 b - 71295/4 + ------ 6615/2 - 2205 b 2 --------------- + ------------------] n 2 n and in Maple-input format: [(3/10-1/5*b)/n+(-3/4+3/10*b)/n^2, (30-20*b)/n+(-110+30*b)/n^2, (6615/2-2205*b) /n+(-71295/4+6615/2*b)/n^2] and in floating-point it is: 0.2236067976 0.6354101964 22.36067976 98.54101964 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 2465.264944 16560.39742 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 3, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 1, d[1] = 1, d[2] = 3 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 2 4 2 2 4 2 2 2 ) A(n, h) t = (-1 - 3 h - h - 4 h (1 + h ) t + 2 h (1 + h ) t / ----- n = 0 6 2 3 10 2 4 5 12 / 2 + 2 h (2 + 3 h ) t - h (1 + h ) t - 2 t h ) / ((h t + 1) / 2 4 8 2 4 2 2 (h t - 1) (t h - 2 t h - 2 t h + 1)) and in Maple input format: (-1-3*h^2-h^4-4*h^2*(1+h^2)*t+2*h^4*(1+h^2)^2*t^2+2*h^6*(2+3*h^2)*t^3-h^10*(1+h ^2)*t^4-2*t^5*h^12)/(h^2*t+1)/(h^2*t-1)/(t^4*h^8-2*t^2*h^4-2*t^2*h^2+1) We now present a statistical analysis 1/2 1/2 6 2 Let, b, be the algebraic number, ---- - ---- 2 2 4 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.5176380910 We have the following proven facts The total number of tilings of the region is asymptotic to: 2 3 n (19/8 + 13/3 b - 5/8 b - 7/6 b ) (1/b) that in floating-point is: n 4.288812654 1.931851650 and in Maple input format: 4.288812654*1.931851650^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: / 2 \ 2 3 | b | 13/3 - 6 b + b + 2 b + |4/3 + ----| n \ 3 / and in Maple-input format: 13/3-6*b+b^2+2*b^3+(4/3+1/3*b^2)*n and in floating point: 1.772855398 + 1.422649731 n The asymptotic expression for the variance, as a function of n is: 2 / 2\ 2 b 3 | 2 b | - 64/9 - ---- - 6 b + 18 b + |4/9 - ----| n 3 \ 9 / and in Maple-input format: -64/9-2/3*b^2-6*b^3+18*b+(4/9-2/9*b^2)*n and in floating point: 1.195537479 + 0.3849001792 n The even alpha coefficients until the, 8, -th , are: 2 2 3 -4 + 2 b -2944 - 6 b - 4185/2 b + 12555/2 b [1, 3 + --------- + ------------------------------------, n 2 n 2 2 3 -60 + 30 b -44112 + 188325/2 b - 90 b - 62775/2 b 15 + ----------- + ----------------------------------------, n 2 n 2 2 3 -840 + 420 b -616476 - 1260 b - 439425 b + 1318275 b 105 + ------------- + -----------------------------------------] n 2 n and in Maple-input format: [1, 3+(-4+2*b^2)/n+(-2944-6*b^2-4185/2*b^3+12555/2*b)/n^2, 15+(-60+30*b^2)/n+(-\ 44112+188325/2*b-90*b^2-62775/2*b^3)/n^2, 105+(-840+420*b^2)/n+(-616476-1260*b^ 2-439425*b^3+1318275*b)/n^2] and in floating-point it is: 3.464101613 13.634188 51.96152420 252.512812 [1, 3 - ----------- + ---------, 15 - ----------- + ----------, n 2 n 2 n n 727.4613388 4627.1794 105 - ----------- + ---------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 4, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 1, d[1] = 2, d[2] = 1 Then infinity ----- 2 2 2 2 7 2 2 \ n -h (2 + h ) + h (2 h + 1) (1 + h ) t - h (1 + h ) t ) A(n, h) t = ------------------------------------------------------- / 3 6 2 3 ----- (h t - 1) (h t - 2 h t - h t + 1) n = 0 and in Maple input format: (-h*(2+h^2)+h^2*(2*h^2+1)*(1+h^2)*t-h^7*(1+h^2)*t^2)/(h^3*t-1)/(h^6*t^2-2*h^3*t -h*t+1) We now present a statistical analysis 1/2 5 Let, b, be the algebraic number, 3/2 - ---- 2 2 whose minimal polynomial is, b - 3 b + 1 and whose floating-point appx. , to, 10, digits is, 0.381966012 We have the following proven facts The total number of tilings of the region is asymptotic to: / 4 b\ n |11/5 - ---| (1/b) \ 5 / that in floating-point is: n 1.894427190 2.618033984 and in Maple input format: 1.894427190*2.618033984^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 6 b / 4 b\ 7/5 + --- + |9/5 + ---| n 5 \ 5 / and in Maple-input format: 7/5+6/5*b+(9/5+4/5*b)*n and in floating point: 1.858359214 + 2.105572810 n The asymptotic expression for the variance, as a function of n is: 28 24 b /24 16 b\ -- - ---- + |-- - ----| n 25 25 \25 25 / and in Maple-input format: 28/25-24/25*b+(24/25-16/25*b)*n and in floating point: 0.7533126285 + 0.7155417523 n The even alpha coefficients until the, 8, -th , are: 23 3 b 39 b -- - --- 19/2 - ---- - 3/2 + b 20 2 13 b - 39/2 2 [1, 3 + --------- + --------, 15 + ----------- + -----------, n 2 n 2 n n -231 + 154 b - 119/4 - 231 b 105 + ------------ + ---------------] n 2 n and in Maple-input format: [1, 3+(-3/2+b)/n+(23/20-3/2*b)/n^2, 15+(13*b-39/2)/n+(19/2-39/2*b)/n^2, 105+(-\ 231+154*b)/n+(-119/4-231*b)/n^2] and in floating-point it is: 1.118033988 0.5770509820 14.53444184 2.051662766 [1, 3 - ----------- + ------------, 15 - ----------- + -----------, n 2 n 2 n n 172.1772342 117.9841488 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 3 b - 3/4 + --- 3/10 - b/5 10 30 - 20 b -110 + 30 b [---------- + -----------, --------- + -----------, n 2 n 2 n n 6615 b - 71295/4 + ------ 6615/2 - 2205 b 2 --------------- + ------------------] n 2 n and in Maple-input format: [(3/10-1/5*b)/n+(-3/4+3/10*b)/n^2, (30-20*b)/n+(-110+30*b)/n^2, (6615/2-2205*b) /n+(-71295/4+6615/2*b)/n^2] and in floating-point it is: 0.2236067976 0.6354101964 22.36067976 98.54101964 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 2465.264944 16560.39742 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 5, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 1, d[1] = 2, d[2] = 2 Then infinity ----- \ n ) A(n, h) t = / ----- n = 0 2 4 2 2 4 2 10 2 2 -1 - 3 h - h + h (1 + h ) (2 h + 4 h + 1) t - h (1 + h ) t ------------------------------------------------------------------ 4 2 8 4 2 (t h - 1) (t h - 2 t h - 4 h t - t + 1) and in Maple input format: (-1-3*h^2-h^4+h^2*(1+h^2)*(2*h^4+4*h^2+1)*t-h^10*(1+h^2)*t^2)/(t*h^4-1)/(t^2*h^ 8-2*t*h^4-4*h^2*t-t+1) We now present a statistical analysis 1/2 3 5 Let, b, be the algebraic number, 7/2 - ------ 2 2 whose minimal polynomial is, b - 7 b + 1 and whose floating-point appx. , to, 10, digits is, 0.145898034 We have the following proven facts The total number of tilings of the region is asymptotic to: / 8 b\ n |11/3 - ---| (1/b) \ 15 / that in floating-point is: n 3.588854382 6.854101955 and in Maple input format: 3.588854382*6.854101955^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 4 b /32 8 b\ 8/5 + --- + |-- + ---| n 5 \15 15 / and in Maple-input format: 8/5+4/5*b+(32/15+8/15*b)*n and in floating point: 1.716718427 + 2.211145618 n The asymptotic expression for the variance, as a function of n is: 16 b /112 32 b\ 8/5 - ---- + |--- - ----| n 25 \75 75 / and in Maple-input format: 8/5-16/25*b+(112/75-32/75*b)*n and in floating point: 1.506625258 + 1.431083505 n The even alpha coefficients until the, 8, -th , are: 13 13 b 91 25 13 b -- - b/4 ---- - -- -- - ---- - 7/12 + b/6 40 6 12 16 4 [1, 3 + ------------ + --------, 15 + --------- + ---------, n 2 n 2 n n 77 b 609 77 b - 539/6 + ---- - --- - ---- 3 16 2 105 + -------------- + ------------] n 2 n and in Maple-input format: [1, 3+(-7/12+1/6*b)/n+(13/40-1/4*b)/n^2, 15+(13/6*b-91/12)/n+(25/16-13/4*b)/n^2 , 105+(-539/6+77/3*b)/n+(-609/16-77/2*b)/n^2] and in floating-point it is: 0.5590169943 0.2885254915 7.267220926 1.088331390 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 86.08861712 43.67957431 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 13 b 10 b 7/60 - ---- - -- + ---- 35/3 - ---- 30 40 20 3 - 165/4 + 5 b [----------- + -----------, ----------- + -------------, n 2 n 2 n n 735 b 2205 b 5145/4 - ----- - 47775/8 + ------ 2 4 -------------- + ------------------] n 2 n and in Maple-input format: [(7/60-1/30*b)/n+(-13/40+1/20*b)/n^2, (35/3-10/3*b)/n+(-165/4+5*b)/n^2, (5145/4 -735/2*b)/n+(-47775/8+2205/4*b)/n^2] and in floating-point it is: 0.1118033989 0.3177050983 11.18033989 40.52050983 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 1232.632472 5891.448709 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 6, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 1, d[1] = 2, d[2] = 3 Then infinity ----- \ n 2 2 ) A(n, h) t = (-h (h + 3) (1 + h ) / ----- n = 0 2 6 2 4 8 + h (2 + 26 h + 12 h + 29 h + 6 h ) t 7 6 2 8 4 2 - h (9 + 65 h + 47 h + 15 h + 85 h ) t 12 2 8 6 4 3 + h (12 + 58 h + 20 h + 80 h + 105 h ) t 17 2 6 4 2 4 - h (1 + h ) (15 h + 35 h + 21 h + 4) t 24 2 2 2 5 33 2 6 / 5 + 2 h (1 + 3 h ) (1 + h ) t - h (1 + h ) t ) / ((t h - 1) / 2 10 3 5 4 20 3 15 3 13 3 11 (t h + 1 - 2 h t - 2 t h ) (t h - 4 t h - 6 t h - 2 t h 2 10 2 8 6 2 5 3 + 6 t h + 12 t h + 5 h t - 4 t h - 6 h t - 2 h t + 1)) and in Maple input format: (-h*(h^2+3)*(1+h^2)+h^2*(2+26*h^6+12*h^2+29*h^4+6*h^8)*t-h^7*(9+65*h^6+47*h^2+ 15*h^8+85*h^4)*t^2+h^12*(12+58*h^2+20*h^8+80*h^6+105*h^4)*t^3-h^17*(1+h^2)*(15* h^6+35*h^4+21*h^2+4)*t^4+2*h^24*(1+3*h^2)*(1+h^2)^2*t^5-h^33*(1+h^2)*t^6)/(t*h^ 5-1)/(t^2*h^10+1-2*h^3*t-2*t*h^5)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-2*t^3*h^11+6* t^2*h^10+12*t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /1471 310 166 2 139 3\ n |---- - --- b + --- b - --- b | (1/b) \210 21 21 210 / that in floating-point is: n 5.57601090844784 9.77063586195681 and in Maple input format: HFloat(5.57601090844783887)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 188 48 b 22 b 9 b /263 44 2 3\ --- + ---- - ----- + ---- + |--- + 32/7 b - -- b + 6/35 b | n 105 7 7 35 \105 21 / and in Maple-input format: 188/105+48/7*b-22/7*b^2+9/35*b^3+(263/105+32/7*b-44/21*b^2+6/35*b^3)*n and in floating point: 2.45964173878839 + 2.95087227085893 n The asymptotic expression for the variance, as a function of n is: 3 2 3212 46 b 328 b 208 b /2944 416 656 2 92 3\ ---- - ----- + ------ - ----- + |---- - --- b + --- b - --- b | n 1575 175 105 35 \1575 105 315 525 / and in Maple-input format: 3212/1575-46/175*b^3+328/105*b^2-208/35*b+(2944/1575-416/105*b+656/315*b^2-92/ 525*b^3)*n and in floating point: 1.46356872276963 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 178709207 368140044 188363694 2 30921315 3 --------- - --------- b + --------- b - -------- b 171955 34391 34391 68782 + ----------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 + ----------------------------------------------------- n 33168240255 342705312240 183068876070 2 30364686759 3 - ----------- + ------------ b - ------------ b + ----------- b 584647 584647 584647 1169294 + -----------------------------------------------------------------, 105 2 n 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b 4913 4913 4913 4913 + --------------------------------------------------------- n 131028594897 1353036161760 716230164780 2 59271740223 3 - ------------ + ------------- b - ------------ b + ----------- b 83521 83521 83521 83521 + -------------------------------------------------------------------] 2 n and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 178709207/171955-368140044/34391*b+188363694/34391*b^2-30921315/68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+36770751/34391*b^3)/n +(-33168240255/584647+342705312240/584647*b-183068876070/584647*b^2+30364686759 /1169294*b^3)/n^2, 105+(-246835536/4913+2637282960/4913*b-2293010520/4913*b^2+ 207323694/4913*b^3)/n+(-131028594897/83521+1353036161760/83521*b-716230164780/ 83521*b^2+59271740223/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.586488850215835 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 9.26248996964857 15 - ---------------- + ----------------, n 2 n 145.184702468171 147.037129145152 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 [--------------------------------------------------- n 18344930292 37872665304 19954468872 2 8247533466 3 - ----------- + ----------- b - ----------- b + ---------- b 2923235 584647 584647 2923235 + --------------------------------------------------------------, 2 n 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b 34391 34391 34391 34391 --------------------------------------------------------- n 269504728560 2779893178800 1445970513600 2 119160971520 3 - ------------ + ------------- b - ------------- b + ------------ b 584647 584647 584647 584647 + ---------------------------------------------------------------------, 2 n 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 180537137515980 1869366969187200 1038486868335000 2 --------------- - ---------------- b + ---------------- b 1419857 1419857 1419857 86898439321470 3\ / 2 - -------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -18344930292/2923235+37872665304/584647*b-19954468872/584647*b^2+8247533466/ 2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-7397956800/34391*b^2+ 668910960/34391*b^3)/n+(-269504728560/584647+2779893178800/584647*b-\ 1445970513600/584647*b^2+119160971520/584647*b^3)/n^2, (-12501125280/4913+ 133972120800/4913*b-116517819600/4913*b^2+10535347620/4913*b^3)/n+( 180537137515980/1419857-1869366969187200/1419857*b+1038486868335000/1419857*b^2 -86898439321470/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.117735852541510 2.49983457428517 14.4632647162786 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 2024.30307778587 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 7, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 1, d[1] = 3, d[2] = 1 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 2 4 2 2 4 2 2 2 ) A(n, h) t = (-1 - 3 h - h - 4 h (1 + h ) t + 2 h (1 + h ) t / ----- n = 0 6 2 3 10 2 4 5 12 / 2 + 2 h (2 + 3 h ) t - h (1 + h ) t - 2 t h ) / ((h t + 1) / 2 4 8 2 4 2 2 (h t - 1) (t h - 2 t h - 2 t h + 1)) and in Maple input format: (-1-3*h^2-h^4-4*h^2*(1+h^2)*t+2*h^4*(1+h^2)^2*t^2+2*h^6*(2+3*h^2)*t^3-h^10*(1+h ^2)*t^4-2*t^5*h^12)/(h^2*t+1)/(h^2*t-1)/(t^4*h^8-2*t^2*h^4-2*t^2*h^2+1) We now present a statistical analysis 1/2 1/2 6 2 Let, b, be the algebraic number, ---- - ---- 2 2 4 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.5176380910 We have the following proven facts The total number of tilings of the region is asymptotic to: 2 3 n (19/8 + 13/3 b - 5/8 b - 7/6 b ) (1/b) that in floating-point is: n 4.288812654 1.931851650 and in Maple input format: 4.288812654*1.931851650^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: / 2 \ 2 3 | b | 13/3 - 6 b + b + 2 b + |4/3 + ----| n \ 3 / and in Maple-input format: 13/3-6*b+b^2+2*b^3+(4/3+1/3*b^2)*n and in floating point: 1.772855398 + 1.422649731 n The asymptotic expression for the variance, as a function of n is: 2 / 2\ 2 b 3 | 2 b | - 64/9 - ---- - 6 b + 18 b + |4/9 - ----| n 3 \ 9 / and in Maple-input format: -64/9-2/3*b^2-6*b^3+18*b+(4/9-2/9*b^2)*n and in floating point: 1.195537479 + 0.3849001792 n The even alpha coefficients until the, 8, -th , are: 2 2 3 -4 + 2 b -2944 - 6 b - 4185/2 b + 12555/2 b [1, 3 + --------- + ------------------------------------, n 2 n 2 2 3 -60 + 30 b -44112 + 188325/2 b - 90 b - 62775/2 b 15 + ----------- + ----------------------------------------, n 2 n 2 2 3 -840 + 420 b -616476 - 1260 b - 439425 b + 1318275 b 105 + ------------- + -----------------------------------------] n 2 n and in Maple-input format: [1, 3+(-4+2*b^2)/n+(-2944-6*b^2-4185/2*b^3+12555/2*b)/n^2, 15+(-60+30*b^2)/n+(-\ 44112+188325/2*b-90*b^2-62775/2*b^3)/n^2, 105+(-840+420*b^2)/n+(-616476-1260*b^ 2-439425*b^3+1318275*b)/n^2] and in floating-point it is: 3.464101613 13.634188 51.96152420 252.512812 [1, 3 - ----------- + ---------, 15 - ----------- + ----------, n 2 n 2 n n 727.4613388 4627.1794 105 - ----------- + ---------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 8, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 1, d[1] = 3, d[2] = 2 Then infinity ----- \ n 2 2 ) A(n, h) t = (-h (h + 3) (1 + h ) / ----- n = 0 2 6 2 4 8 + h (2 + 26 h + 12 h + 29 h + 6 h ) t 7 6 2 8 4 2 - h (9 + 65 h + 47 h + 15 h + 85 h ) t 12 2 8 6 4 3 + h (12 + 58 h + 20 h + 80 h + 105 h ) t 17 2 6 4 2 4 - h (1 + h ) (15 h + 35 h + 21 h + 4) t 24 2 2 2 5 33 2 6 / 5 + 2 h (1 + 3 h ) (1 + h ) t - h (1 + h ) t ) / ((t h - 1) / 2 10 3 5 4 20 3 15 3 13 3 11 (t h + 1 - 2 h t - 2 t h ) (t h - 4 t h - 6 t h - 2 t h 2 10 2 8 6 2 5 3 + 6 t h + 12 t h + 5 h t - 4 t h - 6 h t - 2 h t + 1)) and in Maple input format: (-h*(h^2+3)*(1+h^2)+h^2*(2+26*h^6+12*h^2+29*h^4+6*h^8)*t-h^7*(9+65*h^6+47*h^2+ 15*h^8+85*h^4)*t^2+h^12*(12+58*h^2+20*h^8+80*h^6+105*h^4)*t^3-h^17*(1+h^2)*(15* h^6+35*h^4+21*h^2+4)*t^4+2*h^24*(1+3*h^2)*(1+h^2)^2*t^5-h^33*(1+h^2)*t^6)/(t*h^ 5-1)/(t^2*h^10+1-2*h^3*t-2*t*h^5)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-2*t^3*h^11+6* t^2*h^10+12*t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /1471 310 166 2 139 3\ n |---- - --- b + --- b - --- b | (1/b) \210 21 21 210 / that in floating-point is: n 5.57601090844784 9.77063586195681 and in Maple input format: HFloat(5.57601090844783887)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 188 48 b 22 b 9 b /263 44 2 3\ --- + ---- - ----- + ---- + |--- + 32/7 b - -- b + 6/35 b | n 105 7 7 35 \105 21 / and in Maple-input format: 188/105+48/7*b-22/7*b^2+9/35*b^3+(263/105+32/7*b-44/21*b^2+6/35*b^3)*n and in floating point: 2.45964173878839 + 2.95087227085893 n The asymptotic expression for the variance, as a function of n is: 3 2 3212 46 b 328 b 208 b /2944 416 656 2 92 3\ ---- - ----- + ------ - ----- + |---- - --- b + --- b - --- b | n 1575 175 105 35 \1575 105 315 525 / and in Maple-input format: 3212/1575-46/175*b^3+328/105*b^2-208/35*b+(2944/1575-416/105*b+656/315*b^2-92/ 525*b^3)*n and in floating point: 1.46356872276963 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 178709207 368140044 188363694 2 30921315 3 --------- - --------- b + --------- b - -------- b 171955 34391 34391 68782 + ----------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 + ----------------------------------------------------- n 33168240255 342705312240 183068876070 2 30364686759 3 - ----------- + ------------ b - ------------ b + ----------- b 584647 584647 584647 1169294 + -----------------------------------------------------------------, 105 2 n 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b 4913 4913 4913 4913 + --------------------------------------------------------- n 131028594897 1353036161760 716230164780 2 59271740223 3 - ------------ + ------------- b - ------------ b + ----------- b 83521 83521 83521 83521 + -------------------------------------------------------------------] 2 n and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 178709207/171955-368140044/34391*b+188363694/34391*b^2-30921315/68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+36770751/34391*b^3)/n +(-33168240255/584647+342705312240/584647*b-183068876070/584647*b^2+30364686759 /1169294*b^3)/n^2, 105+(-246835536/4913+2637282960/4913*b-2293010520/4913*b^2+ 207323694/4913*b^3)/n+(-131028594897/83521+1353036161760/83521*b-716230164780/ 83521*b^2+59271740223/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.586488850215835 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 9.26248996964857 15 - ---------------- + ----------------, n 2 n 145.184702468171 147.037129145152 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 [--------------------------------------------------- n 18344930292 37872665304 19954468872 2 8247533466 3 - ----------- + ----------- b - ----------- b + ---------- b 2923235 584647 584647 2923235 + --------------------------------------------------------------, 2 n 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b 34391 34391 34391 34391 --------------------------------------------------------- n 269504728560 2779893178800 1445970513600 2 119160971520 3 - ------------ + ------------- b - ------------- b + ------------ b 584647 584647 584647 584647 + ---------------------------------------------------------------------, 2 n 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 180537137515980 1869366969187200 1038486868335000 2 --------------- - ---------------- b + ---------------- b 1419857 1419857 1419857 86898439321470 3\ / 2 - -------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -18344930292/2923235+37872665304/584647*b-19954468872/584647*b^2+8247533466/ 2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-7397956800/34391*b^2+ 668910960/34391*b^3)/n+(-269504728560/584647+2779893178800/584647*b-\ 1445970513600/584647*b^2+119160971520/584647*b^3)/n^2, (-12501125280/4913+ 133972120800/4913*b-116517819600/4913*b^2+10535347620/4913*b^3)/n+( 180537137515980/1419857-1869366969187200/1419857*b+1038486868335000/1419857*b^2 -86898439321470/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.117735852541510 2.49983457428517 14.4632647162786 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 2024.30307778587 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 9, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 1, d[1] = 3, d[2] = 3 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 2 6 4 2 2 2 ) A(n, h) t = (-1 - 6 h - h - 5 h - 8 h (1 + h ) t / ----- n = 0 4 2 6 4 2 2 + 2 h (2 h + 1) (h + 5 h + 6 h + 3) t 6 2 4 6 3 + 2 h (8 + 28 h + 32 h + 11 h ) t 10 6 4 2 8 4 - h (30 h + 50 h + 33 h + 8 + 6 h ) t 12 2 4 6 5 - 2 h (8 + 28 h + 30 h + 11 h ) t 18 2 2 2 6 20 4 2 7 + 2 h (2 h + 1) (2 + h ) (1 + h ) t + 2 h (4 + 5 h + 10 h ) t 28 2 8 9 30 / 3 3 - h (1 + h ) t - 2 t h ) / ((h t + 1) (h t - 1) / 6 2 3 6 2 3 (h t + 1 + 2 h t + 2 h t) (h t + 1 - 2 h t - 2 h t) 4 12 6 2 2 4 (t h - 2 h t - 2 t h + 1)) and in Maple input format: (-1-6*h^2-h^6-5*h^4-8*h^2*(1+h^2)^2*t+2*h^4*(2*h^2+1)*(h^6+5*h^4+6*h^2+3)*t^2+2 *h^6*(8+28*h^2+32*h^4+11*h^6)*t^3-h^10*(30*h^6+50*h^4+33*h^2+8+6*h^8)*t^4-2*h^ 12*(8+28*h^2+30*h^4+11*h^6)*t^5+2*h^18*(2*h^2+1)*(2+h^2)*(1+h^2)*t^6+2*h^20*(4+ 5*h^4+10*h^2)*t^7-h^28*(1+h^2)*t^8-2*t^9*h^30)/(h^3*t+1)/(h^3*t-1)/(h^6*t^2+1+2 *h*t+2*h^3*t)/(h^6*t^2+1-2*h*t-2*h^3*t)/(t^4*h^12-2*h^6*t^2-2*t^2*h^4+1) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 2 - 3 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.267949192 We have the following proven facts The total number of tilings of the region is asymptotic to: / 45 b\ n |21/2 - ----| (1/b) \ 16 / that in floating-point is: n 9.746392898 3.732050814 and in Maple input format: 9.746392898*3.732050814^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: / 2 b\ 2 + 2 b + |5/3 + ---| n \ 3 / and in Maple-input format: 2+2*b+(5/3+2/3*b)*n and in floating point: 2.535898384 + 1.845299462 n The asymptotic expression for the variance, as a function of n is: 4 b / 4 b\ 20/9 - --- + |8/9 - ---| n 3 \ 9 / and in Maple-input format: 20/9-4/3*b+(8/9-4/9*b)*n and in floating point: 1.864956633 + 0.7698003591 n The even alpha coefficients until the, 8, -th , are: b - 2 23/4 - 3 b 15 b - 30 393/4 - 45 b [1, 3 + ----- + ----------, 15 + --------- + ------------, n 2 n 2 n n -420 + 210 b 3297/2 - 630 b 105 + ------------ + --------------] n 2 n and in Maple-input format: [1, 3+(b-2)/n+(23/4-3*b)/n^2, 15+(15*b-30)/n+(393/4-45*b)/n^2, 105+(-420+210*b) /n+(3297/2-630*b)/n^2] and in floating-point it is: 1.732050808 4.946152424 25.98076212 86.19228636 [1, 3 - ----------- + -----------, 15 - ----------- + -----------, n 2 n 2 n n 363.7306697 1479.692009 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 10, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 2, d[1] = 1, d[2] = 1 Then infinity ----- 2 2 2 2 4 2 2 \ n -2 h - 1 + h (2 + h ) (1 + h ) t - h (1 + h ) t ) A(n, h) t = --------------------------------------------------- / 2 4 2 4 2 ----- (h t - 1) (-t h + t h - 2 h t + 1) n = 0 and in Maple input format: (-2*h^2-1+h^2*(2+h^2)*(1+h^2)*t-h^4*(1+h^2)*t^2)/(h^2*t-1)/(-t*h^4+t^2*h^4-2*h^ 2*t+1) We now present a statistical analysis 1/2 5 Let, b, be the algebraic number, 3/2 - ---- 2 2 whose minimal polynomial is, b - 3 b + 1 and whose floating-point appx. , to, 10, digits is, 0.381966012 We have the following proven facts The total number of tilings of the region is asymptotic to: / 4 b\ n |11/5 - ---| (1/b) \ 5 / that in floating-point is: n 1.894427190 2.618033984 and in Maple input format: 1.894427190*2.618033984^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 6 b / 4 b\ 8/5 - --- + |16/5 - ---| n 5 \ 5 / and in Maple-input format: 8/5-6/5*b+(16/5-4/5*b)*n and in floating point: 1.141640786 + 2.894427190 n The asymptotic expression for the variance, as a function of n is: 28 24 b /24 16 b\ -- - ---- + |-- - ----| n 25 25 \25 25 / and in Maple-input format: 28/25-24/25*b+(24/25-16/25*b)*n and in floating point: 0.7533126285 + 0.7155417523 n The even alpha coefficients until the, 8, -th , are: 23 3 b 39 b -- - --- 19/2 - ---- - 3/2 + b 20 2 13 b - 39/2 2 [1, 3 + --------- + --------, 15 + ----------- + -----------, n 2 n 2 n n -231 + 154 b - 119/4 - 231 b 105 + ------------ + ---------------] n 2 n and in Maple-input format: [1, 3+(-3/2+b)/n+(23/20-3/2*b)/n^2, 15+(13*b-39/2)/n+(19/2-39/2*b)/n^2, 105+(-\ 231+154*b)/n+(-119/4-231*b)/n^2] and in floating-point it is: 1.118033988 0.5770509820 14.53444184 2.051662766 [1, 3 - ----------- + ------------, 15 - ----------- + -----------, n 2 n 2 n n 172.1772342 117.9841488 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 3 b - 3/4 + --- 3/10 - b/5 10 30 - 20 b -110 + 30 b [---------- + -----------, --------- + -----------, n 2 n 2 n n 6615 b - 71295/4 + ------ 6615/2 - 2205 b 2 --------------- + ------------------] n 2 n and in Maple-input format: [(3/10-1/5*b)/n+(-3/4+3/10*b)/n^2, (30-20*b)/n+(-110+30*b)/n^2, (6615/2-2205*b) /n+(-71295/4+6615/2*b)/n^2] and in floating-point it is: 0.2236067976 0.6354101964 22.36067976 98.54101964 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 2465.264944 16560.39742 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 11, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 2, d[1] = 1, d[2] = 3 Then infinity ----- \ n 2 4 ) A(n, h) t = (1 + 6 h + 4 h / ----- n = 0 2 2 6 4 8 - h (27 h + 4 + 38 h + 51 h + 9 h ) t 4 4 2 6 8 10 12 2 + h (110 h + 4 + 36 h + 163 h + 127 h + 48 h + 7 h ) t 8 10 8 6 4 2 14 12 3 - h (80 h + 186 h + 250 h + 192 h + 76 h + 12 + 2 h + 19 h ) t 12 8 10 2 4 6 14 12 4 + h (154 h + 8 + 63 h + 52 h + 141 h + 201 h + h + 13 h ) t 18 2 8 6 4 2 5 - h (1 + h ) (3 h + 20 h + 43 h + 39 h + 12) t 24 2 4 2 6 30 2 7 / + h (1 + h ) (6 + 3 h + 10 h ) t - h (1 + h ) t ) / ( / 2 8 4 6 2 8 2 4 4 16 (t h + 1 - 2 t h - t h ) (t h + 1 - 2 h t - 2 t h ) (t h 3 14 3 12 2 12 3 10 2 10 2 8 6 2 - 2 t h - 6 t h + t h - 4 t h + 4 t h + 6 t h + 8 h t 6 2 4 4 2 - 2 t h + 4 t h - 6 t h - 4 h t + 1)) and in Maple input format: (1+6*h^2+4*h^4-h^2*(27*h^2+4+38*h^6+51*h^4+9*h^8)*t+h^4*(110*h^4+4+36*h^2+163*h ^6+127*h^8+48*h^10+7*h^12)*t^2-h^8*(80*h^10+186*h^8+250*h^6+192*h^4+76*h^2+12+2 *h^14+19*h^12)*t^3+h^12*(154*h^8+8+63*h^10+52*h^2+141*h^4+201*h^6+h^14+13*h^12) *t^4-h^18*(1+h^2)*(3*h^8+20*h^6+43*h^4+39*h^2+12)*t^5+h^24*(1+h^2)*(6+3*h^4+10* h^2)*t^6-h^30*(1+h^2)*t^7)/(t^2*h^8+1-2*t*h^4-t*h^6)/(t^2*h^8+1-2*h^2*t-2*t*h^4 )/(t^4*h^16-2*t^3*h^14-6*t^3*h^12+t^2*h^12-4*t^3*h^10+4*t^2*h^10+6*t^2*h^8+8*h^ 6*t^2-2*t*h^6+4*t^2*h^4-6*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /482 172 2 26 3\ 23/3 - ----- + ------ - ----- + |--- - 64/7 b + --- b - -- b | n 15 15 15 \105 21 35 / and in Maple-input format: 23/3-842/15*b+662/15*b^2-59/15*b^3+(482/105-64/7*b+172/21*b^2-26/35*b^3)*n and in floating point: 2.37964132686805 + 3.73972665214447 n The asymptotic expression for the variance, as a function of n is: 3 2 794308 b 25936 52546 b 593848 b /2944 416 656 2 92 3\ - -------- + ----- - -------- + --------- + |---- - --- b + --- b - --- b | n 1575 525 1575 1575 \1575 105 315 525 / and in Maple-input format: -794308/1575*b+25936/525-52546/1575*b^3+593848/1575*b^2+(2944/1575-416/105*b+ 656/315*b^2-92/525*b^3)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 12, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 2, d[1] = 2, d[2] = 2 Then infinity ----- \ n 2 4 ) A(n, h) t = (-1 - 6 h - 4 h / ----- n = 0 8 10 6 4 2 + (1 + 17 h + 3 h + 32 h + 27 h + 9 h ) t 2 2 12 10 8 6 4 2 2 - h (1 + h ) (h + 11 h + 32 h + 49 h + 34 h + 10 h + 1) t 6 2 4 10 8 6 4 2 3 + h (1 + 3 h + h ) (3 h + 9 h + 21 h + 28 h + 14 h + 2) t 10 10 4 12 14 8 2 6 4 - h (39 h + 31 h + 11 h + h + 1 + 64 h + 9 h + 57 h ) t 18 4 2 2 2 5 26 2 6 / 4 + h (2 h + 7 h + 2) (1 + h ) t - h (1 + h ) t ) / ((t h - 1) / 2 8 4 6 4 16 3 14 3 12 2 12 (t h + 1 - 2 t h - t h ) (t h - 2 t h - 8 t h + t h 3 10 2 10 3 8 2 8 6 2 6 2 4 - 9 t h + 4 t h - 2 t h + 6 t h + 16 h t - 2 t h + 20 t h 4 2 2 2 2 - 8 t h + 8 t h - 9 h t + t - 2 t + 1)) and in Maple input format: (-1-6*h^2-4*h^4+(1+17*h^8+3*h^10+32*h^6+27*h^4+9*h^2)*t-h^2*(1+h^2)*(h^12+11*h^ 10+32*h^8+49*h^6+34*h^4+10*h^2+1)*t^2+h^6*(1+3*h^2+h^4)*(3*h^10+9*h^8+21*h^6+28 *h^4+14*h^2+2)*t^3-h^10*(39*h^10+31*h^4+11*h^12+h^14+1+64*h^8+9*h^2+57*h^6)*t^4 +h^18*(2*h^4+7*h^2+2)*(1+h^2)^2*t^5-h^26*(1+h^2)*t^6)/(t*h^4-1)/(t^2*h^8+1-2*t* h^4-t*h^6)/(t^4*h^16-2*t^3*h^14-8*t^3*h^12+t^2*h^12-9*t^3*h^10+4*t^2*h^10-2*t^3 *h^8+6*t^2*h^8+16*h^6*t^2-2*t*h^6+20*t^2*h^4-8*t*h^4+8*t^2*h^2-9*h^2*t+t^2-2*t+ 1) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 9 - 4 5 2 whose minimal polynomial is, b - 18 b + 1 and whose floating-point appx. , to, 10, digits is, 0.055728092 We have the following proven facts The total number of tilings of the region is asymptotic to: /521 29 b\ n |--- - ----| (1/b) \50 50 / that in floating-point is: n 10.38767771 17.94427127 and in Maple input format: 10.38767771*17.94427127^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 57 3 b /31 b \ -- + --- + |-- + ----| n 20 20 \10 10 / and in Maple-input format: 57/20+3/20*b+(31/10+1/10*b)*n and in floating point: 2.858359214 + 3.105572809 n The asymptotic expression for the variance, as a function of n is: 49 7 b /54 6 b\ -- - --- + |-- - ---| n 25 25 \25 25 / and in Maple-input format: 49/25-7/25*b+(54/25-6/25*b)*n and in floating point: 1.944396134 + 2.146625258 n The even alpha coefficients until the, 8, -th , are: b 103 7 b 133 b 133 439 313 b - 3/8 + ---- --- - --- ----- - --- --- - ----- 24 360 144 216 24 108 432 [1, 3 + ------------ + ---------, 15 + ----------- + -----------, n 2 n 2 n n 917 917 b 4648 2177 b - --- + ----- ---- - ------ 12 108 81 216 105 + ------------- + -------------] n 2 n and in Maple-input format: [1, 3+(-3/8+1/24*b)/n+(103/360-7/144*b)/n^2, 15+(133/216*b-133/24)/n+(439/108-\ 313/432*b)/n^2, 105+(-917/12+917/108*b)/n+(4648/81-2177/216*b)/n^2] and in floating-point it is: 0.3726779962 0.2834021066 5.507352610 4.024437748 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 75.94349389 56.82104912 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 43 b 5 b 785 5 b 1/120 - ---- - ---- + ---- 5/6 - --- - --- + --- 1080 2160 2160 54 324 108 [------------ + -------------, --------- + -----------, n 2 n 2 n n 245 b 145775 245 b 735/8 - ----- - ------ + ----- 24 432 48 ------------- + ----------------] n 2 n and in Maple-input format: [(1/120-1/1080*b)/n+(-43/2160+1/2160*b)/n^2, (5/6-5/54*b)/n+(-785/324+5/108*b)/ n^2, (735/8-245/24*b)/n+(-145775/432+245/48*b)/n^2] and in floating-point it is: 0.008281733248 0.01988160737 0.8281733248 2.420259502 [-------------- - -------------, ------------ - -----------, n 2 n 2 n n 91.30610906 337.1576841 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 13, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 2, d[1] = 3, d[2] = 1 Then infinity ----- \ n 2 4 ) A(n, h) t = (1 + 6 h + 4 h / ----- n = 0 2 2 6 4 8 - h (27 h + 4 + 38 h + 51 h + 9 h ) t 4 4 2 6 8 10 12 2 + h (110 h + 4 + 36 h + 163 h + 127 h + 48 h + 7 h ) t 8 10 8 6 4 2 14 12 3 - h (80 h + 186 h + 250 h + 192 h + 76 h + 12 + 2 h + 19 h ) t 12 8 10 2 4 6 14 12 4 + h (154 h + 8 + 63 h + 52 h + 141 h + 201 h + h + 13 h ) t 18 2 8 6 4 2 5 - h (1 + h ) (3 h + 20 h + 43 h + 39 h + 12) t 24 2 4 2 6 30 2 7 / + h (1 + h ) (6 + 3 h + 10 h ) t - h (1 + h ) t ) / ( / 2 8 4 6 2 8 2 4 4 16 (t h + 1 - 2 t h - t h ) (t h + 1 - 2 h t - 2 t h ) (t h 3 14 3 12 2 12 3 10 2 10 2 8 6 2 - 2 t h - 6 t h + t h - 4 t h + 4 t h + 6 t h + 8 h t 6 2 4 4 2 - 2 t h + 4 t h - 6 t h - 4 h t + 1)) and in Maple input format: (1+6*h^2+4*h^4-h^2*(27*h^2+4+38*h^6+51*h^4+9*h^8)*t+h^4*(110*h^4+4+36*h^2+163*h ^6+127*h^8+48*h^10+7*h^12)*t^2-h^8*(80*h^10+186*h^8+250*h^6+192*h^4+76*h^2+12+2 *h^14+19*h^12)*t^3+h^12*(154*h^8+8+63*h^10+52*h^2+141*h^4+201*h^6+h^14+13*h^12) *t^4-h^18*(1+h^2)*(3*h^8+20*h^6+43*h^4+39*h^2+12)*t^5+h^24*(1+h^2)*(6+3*h^4+10* h^2)*t^6-h^30*(1+h^2)*t^7)/(t^2*h^8+1-2*t*h^4-t*h^6)/(t^2*h^8+1-2*h^2*t-2*t*h^4 )/(t^4*h^16-2*t^3*h^14-6*t^3*h^12+t^2*h^12-4*t^3*h^10+4*t^2*h^10+6*t^2*h^8+8*h^ 6*t^2-2*t*h^6+4*t^2*h^4-6*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /482 172 2 26 3\ 23/3 - ----- + ------ - ----- + |--- - 64/7 b + --- b - -- b | n 15 15 15 \105 21 35 / and in Maple-input format: 23/3-842/15*b+662/15*b^2-59/15*b^3+(482/105-64/7*b+172/21*b^2-26/35*b^3)*n and in floating point: 2.37964132686805 + 3.73972665214447 n The asymptotic expression for the variance, as a function of n is: 3 2 25936 52546 b 593848 b 794308 b /2944 416 656 2 92 3\ ----- - -------- + --------- - -------- + |---- - --- b + --- b - --- b | n 525 1575 1575 1575 \1575 105 315 525 / and in Maple-input format: 25936/525-52546/1575*b^3+593848/1575*b^2-794308/1575*b+(2944/1575-416/105*b+656 /315*b^2-92/525*b^3)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 14, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 2, d[1] = 3, d[2] = 3 Then infinity ----- \ n 2 6 4 ) A(n, h) t = (-1 - 12 h - 8 h - 20 h / ----- n = 0 2 8 6 2 10 12 4 4 + h (505 h + 631 h + 108 h + 200 h + 8 + 31 h + 394 h ) t - h ( 16 6 12 14 4 2 18 493 h + 4336 h + 4967 h + 2069 h + 1532 h + 272 h + 51 h + 16 10 8 2 8 16 14 10 4 + 7466 h + 7185 h ) t + h (10184 h + 23435 h + 45672 h + 7664 h 20 22 12 2 6 8 + 128 + 548 h + 45 h + 38592 h + 1536 h + 21448 h + 38044 h 18 3 12 14 2 4 26 + 3027 h ) t - h (110774 h + 2304 h + 12720 h + 21 h + 192 6 10 24 12 22 8 + 41984 h + 135976 h + 294 h + 144010 h + 1977 h + 91060 h 16 18 20 4 18 12 8 + 63122 h + 26970 h + 8605 h ) t + h (247044 h + 183520 h 6 20 22 28 24 2 10 + 95936 h + 14242 h + 3399 h + 4 h + 579 h + 6912 h + 248584 h 26 16 4 18 14 5 24 + 66 h + 640 + 103394 h + 33408 h + 44161 h + 183770 h ) t - h 28 24 2 22 4 14 6 (3 h + 609 h + 7552 h + 3478 h + 36080 h + 186218 h + 102016 h 12 20 8 18 10 + 251318 h + 704 + 14336 h + 191780 h + 44545 h + 255632 h 26 16 6 30 10 4 8 + 66 h + 104673 h ) t + h (150780 h + 256 + 16384 h + 105672 h 28 2 26 20 22 18 12 + h + 3072 h + 26 h + 9210 h + 2123 h + 28374 h + 154332 h 24 6 16 14 7 38 16 + 314 h + 51376 h + 65746 h + 116160 h ) t - h (11521 h 18 12 2 24 14 4 + 3495 h + 256 + 44142 h + 2576 h + 5 h + 26497 h + 11344 h 10 22 6 8 20 8 46 2 + 53933 h + 85 h + 28860 h + 47480 h + 700 h ) t + h (1 + h ) ( 18 16 14 12 10 8 6 10 h + 120 h + 653 h + 2092 h + 4293 h + 5622 h + 4540 h 4 2 9 + 2184 h + 576 h + 64) t 56 8 6 4 2 2 3 10 - h (10 h + 65 h + 161 h + 160 h + 48) (1 + h ) t 66 2 2 2 2 11 76 2 12 / + h (5 h + 3) (1 + h ) (2 + h ) t - h (1 + h ) t ) / ( / 6 2 12 6 8 (t h - 1) (t h + 1 - 2 t h - t h ) 2 12 2 4 6 8 4 2 8 (t h + 1 - 4 h t - 8 t h - 2 t h ) (1 - 2 t h - 4 t h + 4 t h 4 24 2 10 2 14 2 16 3 20 3 18 3 16 + t h + 8 t h + 4 t h + t h - 2 t h - 6 t h - 4 t h 6 2 12 8 4 2 8 4 24 2 - 6 t h + 6 t h ) (1 - 2 t h - 20 t h + 80 t h + t h - 8 h t 2 4 6 2 2 10 2 14 2 16 3 20 + 16 t h + 64 h t + 32 t h + 4 t h + t h - 2 t h 3 18 3 16 6 2 12 3 14 - 12 t h - 20 t h - 12 t h + 6 t h - 8 t h )) and in Maple input format: (-1-12*h^2-8*h^6-20*h^4+h^2*(505*h^8+631*h^6+108*h^2+200*h^10+8+31*h^12+394*h^4 )*t-h^4*(493*h^16+4336*h^6+4967*h^12+2069*h^14+1532*h^4+272*h^2+51*h^18+16+7466 *h^10+7185*h^8)*t^2+h^8*(10184*h^16+23435*h^14+45672*h^10+7664*h^4+128+548*h^20 +45*h^22+38592*h^12+1536*h^2+21448*h^6+38044*h^8+3027*h^18)*t^3-h^12*(110774*h^ 14+2304*h^2+12720*h^4+21*h^26+192+41984*h^6+135976*h^10+294*h^24+144010*h^12+ 1977*h^22+91060*h^8+63122*h^16+26970*h^18+8605*h^20)*t^4+h^18*(247044*h^12+ 183520*h^8+95936*h^6+14242*h^20+3399*h^22+4*h^28+579*h^24+6912*h^2+248584*h^10+ 66*h^26+640+103394*h^16+33408*h^4+44161*h^18+183770*h^14)*t^5-h^24*(3*h^28+609* h^24+7552*h^2+3478*h^22+36080*h^4+186218*h^14+102016*h^6+251318*h^12+704+14336* h^20+191780*h^8+44545*h^18+255632*h^10+66*h^26+104673*h^16)*t^6+h^30*(150780*h^ 10+256+16384*h^4+105672*h^8+h^28+3072*h^2+26*h^26+9210*h^20+2123*h^22+28374*h^ 18+154332*h^12+314*h^24+51376*h^6+65746*h^16+116160*h^14)*t^7-h^38*(11521*h^16+ 3495*h^18+256+44142*h^12+2576*h^2+5*h^24+26497*h^14+11344*h^4+53933*h^10+85*h^ 22+28860*h^6+47480*h^8+700*h^20)*t^8+h^46*(1+h^2)*(10*h^18+120*h^16+653*h^14+ 2092*h^12+4293*h^10+5622*h^8+4540*h^6+2184*h^4+576*h^2+64)*t^9-h^56*(10*h^8+65* h^6+161*h^4+160*h^2+48)*(1+h^2)^3*t^10+h^66*(5*h^2+3)*(1+h^2)*(2+h^2)^2*t^11-h^ 76*(1+h^2)*t^12)/(t*h^6-1)/(t^2*h^12+1-2*t*h^6-t*h^8)/(t^2*h^12+1-4*h^2*t-8*t*h ^4-2*t*h^6)/(1-2*t*h^8-4*t*h^4+4*t^2*h^8+t^4*h^24+8*t^2*h^10+4*t^2*h^14+t^2*h^ 16-2*t^3*h^20-6*t^3*h^18-4*t^3*h^16-6*t*h^6+6*t^2*h^12)/(1-2*t*h^8-20*t*h^4+80* t^2*h^8+t^4*h^24-8*h^2*t+16*t^2*h^4+64*h^6*t^2+32*t^2*h^10+4*t^2*h^14+t^2*h^16-\ 2*t^3*h^20-12*t^3*h^18-20*t^3*h^16-12*t*h^6+6*t^2*h^12-8*t^3*h^14) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /598897 4419759 2757137 2 65689 3\ n |------ - ------- b + ------- b - ----- b | (1/b) \22440 29920 89760 89760 / that in floating-point is: n 22.6608598731619 36.4645094590774 and in Maple input format: HFloat(22.6608598731618898)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 273 41267 b 8917 b 427 b /13757 2192 1516 2 61 3\ --- - ------- + ------- - ------ + |----- - ---- b + ---- b - --- b | n 34 255 255 510 \2805 187 561 935 / and in Maple-input format: 273/34-41267/255*b+8917/255*b^2-427/510*b^3+(13757/2805-2192/187*b+1516/561*b^2 -61/935*b^3)*n and in floating point: 3.61764186686368 + 4.58502611425567 n The asymptotic expression for the variance, as a function of n is: 2 3 15412414 b 2073113 368597 b 71958134 b ----------- + ------- - --------- - ---------- 42075 42075 42075 42075 /105574 26128 15728 2 622 3\ + |------ - ----- b + ----- b - ----- b | n \42075 2805 8415 14025 / and in Maple-input format: 15412414/42075*b^2+2073113/42075-368597/42075*b^3-71958134/42075*b+(105574/ 42075-26128/2805*b+15728/8415*b^2-622/14025*b^3)*n and in floating point: 2.64580188025536 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 / [1, 3 + ----------------------------------------------- + | n \ 188873229085817 346388310500635 74219199907805 2 7100271025879 3\ --------------- - --------------- b + -------------- b - ------------- b | 3520058080 176002904 176002904 704011616 / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b / 2 25143272 22000363 22000363 176002904 / / n , 15 + ------------------------------------------------------- + | / n \ 54764097096363603 1004357290652248941 215199548380302861 2 ----------------- - ------------------- b + ------------------ b 68993138368 34496569184 34496569184 10293664984842231 3\ / 2 - ----------------- b | / n , 105 68993138368 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 53999462889597927 990332082992233677 212194469088606117 2 ----------------- - ------------------ b + ------------------ b 4928081312 2464040656 2464040656 1449988998989361 3\ / 2 - ---------------- b | / n ] 704011616 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(188873229085817/3520058080-346388310500635/176002904*b+74219199907805/ 176002904*b^2-7100271025879/704011616*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 54764097096363603/68993138368-1004357290652248941/34496569184*b+ 215199548380302861/34496569184*b^2-10293664984842231/68993138368*b^3)/n^2, 105+ (-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752*b^3) /n+(53999462889597927/4928081312-990332082992233677/2464040656*b+ 212194469088606117/2464040656*b^2-1449988998989361/704011616*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.657354281434682 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 10.6712005395274 15 - ---------------- + ----------------, n 2 n 106.402545323854 169.933484072189 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 382233603627279 [------------------------------------------------------ + |- --------------- n \ 344965691840 7010059103755707 1502010544350447 2 71845800648927 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 172482845920 172482845920 344965691840 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 955527986765295 17524112293071255 3754811065899105 2 - --------------- + ----------------- b - ---------------- b 8624142296 4312071148 4312071148 179604249480555 3\ / 2 + --------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 842699705777392635 15454832326232571135 473063578968942405 2 - ------------------ + -------------------- b - ------------------ b 68993138368 34496569184 4928081312 158396742930401235 3\ / 2 + ------------------ b | / n ] 68993138368 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-382233603627279/344965691840+7010059103755707/172482845920* b-1502010544350447/172482845920*b^2+71845800648927/344965691840*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-955527986765295/8624142296+17524112293071255/4312071148*b-\ 3754811065899105/4312071148*b^2+179604249480555/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-842699705777392635/68993138368+15454832326232571135/ 34496569184*b-473063578968942405/4928081312*b^2+158396742930401235/68993138368* b^3)/n^2] and in floating-point it is: 0.00714279180461598 0.0135674876180141 0.714279180461598 1.40858596551540 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 188.117654428738 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 15, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 3, d[1] = 1, d[2] = 1 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 2 4 3 2 2 2 2 2 ) A(n, h) t = (-1 - 3 h - h - 4 h (1 + h ) t + 2 h (1 + h ) t / ----- n = 0 5 2 3 4 2 4 5 7 / + 2 h (3 + 2 h ) t - h (1 + h ) t - 2 t h ) / ((h t + 1) (h t - 1) / 4 4 2 4 2 2 (t h - 2 t h - 2 t h + 1)) and in Maple input format: (-1-3*h^2-h^4-4*h^3*(1+h^2)*t+2*h^2*(1+h^2)^2*t^2+2*h^5*(3+2*h^2)*t^3-h^4*(1+h^ 2)*t^4-2*t^5*h^7)/(h*t+1)/(h*t-1)/(t^4*h^4-2*t^2*h^4-2*t^2*h^2+1) We now present a statistical analysis 1/2 1/2 6 2 Let, b, be the algebraic number, ---- - ---- 2 2 4 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.5176380910 We have the following proven facts The total number of tilings of the region is asymptotic to: 2 3 n (19/8 + 13/3 b - 5/8 b - 7/6 b ) (1/b) that in floating-point is: n 4.288812654 1.931851650 and in Maple input format: 4.288812654*1.931851650^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: / 2 \ 2 3 | b | - 1/3 + 6 b - b - 2 b + |5/3 - ----| n \ 3 / and in Maple-input format: -1/3+6*b-b^2-2*b^3+(5/3-1/3*b^2)*n and in floating point: 2.227144602 + 1.577350269 n The asymptotic expression for the variance, as a function of n is: 2 / 2\ 2 b 3 | 2 b | - 64/9 - ---- - 6 b + 18 b + |4/9 - ----| n 3 \ 9 / and in Maple-input format: -64/9-2/3*b^2-6*b^3+18*b+(4/9-2/9*b^2)*n and in floating point: 1.195537479 + 0.3849001792 n The even alpha coefficients until the, 8, -th , are: 2 2 3 -4 + 2 b -2944 - 6 b - 4185/2 b + 12555/2 b [1, 3 + --------- + ------------------------------------, n 2 n 2 2 3 -60 + 30 b -44112 + 188325/2 b - 90 b - 62775/2 b 15 + ----------- + ----------------------------------------, n 2 n 2 2 3 -840 + 420 b -616476 - 1260 b - 439425 b + 1318275 b 105 + ------------- + -----------------------------------------] n 2 n and in Maple-input format: [1, 3+(-4+2*b^2)/n+(-2944-6*b^2-4185/2*b^3+12555/2*b)/n^2, 15+(-60+30*b^2)/n+(-\ 44112+188325/2*b-90*b^2-62775/2*b^3)/n^2, 105+(-840+420*b^2)/n+(-616476-1260*b^ 2-439425*b^3+1318275*b)/n^2] and in floating-point it is: 3.464101613 13.634188 51.96152420 252.512812 [1, 3 - ----------- + ---------, 15 - ----------- + ----------, n 2 n 2 n n 727.4613388 4627.1794 105 - ----------- + ---------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 16, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 3, d[1] = 1, d[2] = 2 Then infinity ----- \ n 2 4 2 ) A(n, h) t = (h (h + 6 h + 4) / ----- n = 0 3 4 2 6 8 - h (9 + 51 h + 38 h + 27 h + 4 h ) t 4 2 6 10 4 8 12 2 + h (48 h + 163 h + 7 + 36 h + 127 h + 110 h + 4 h ) t 5 8 12 2 14 6 4 10 3 - h (250 h + 76 h + 19 h + 12 h + 186 h + 80 h + 192 h + 2) t 8 2 14 6 10 8 12 4 4 + h (13 h + 8 h + 154 h + 1 + 141 h + 201 h + 52 h + 63 h ) t 13 2 8 6 4 2 5 - h (1 + h ) (12 h + 39 h + 43 h + 20 h + 3) t 18 2 2 4 6 23 2 7 / + h (1 + h ) (3 + 10 h + 6 h ) t - h (1 + h ) t ) / ( / 6 2 3 6 2 3 5 4 12 3 11 (h t - 2 h t - h t + 1) (h t + 1 - 2 h t - 2 t h ) (t h - 4 t h 2 10 3 9 2 8 3 7 6 2 5 2 4 + 4 t h - 6 t h + 8 t h - 2 t h + 6 h t - 4 t h + 4 t h 3 2 2 - 6 h t + t h - 2 h t + 1)) and in Maple input format: (h^2*(h^4+6*h^2+4)-h^3*(9+51*h^4+38*h^2+27*h^6+4*h^8)*t+h^4*(48*h^2+163*h^6+7+ 36*h^10+127*h^4+110*h^8+4*h^12)*t^2-h^5*(250*h^8+76*h^12+19*h^2+12*h^14+186*h^6 +80*h^4+192*h^10+2)*t^3+h^8*(13*h^2+8*h^14+154*h^6+1+141*h^10+201*h^8+52*h^12+ 63*h^4)*t^4-h^13*(1+h^2)*(12*h^8+39*h^6+43*h^4+20*h^2+3)*t^5+h^18*(1+h^2)*(3+10 *h^2+6*h^4)*t^6-h^23*(1+h^2)*t^7)/(h^6*t^2-2*h^3*t-h*t+1)/(h^6*t^2+1-2*h^3*t-2* t*h^5)/(t^4*h^12-4*t^3*h^11+4*t^2*h^10-6*t^3*h^9+8*t^2*h^8-2*t^3*h^7+6*h^6*t^2-\ 4*t*h^5+4*t^2*h^4-6*h^3*t+t^2*h^2-2*h*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /253 172 2 26 3\ - 5/3 + ----- - ------ + ----- + |--- + 64/7 b - --- b + -- b | n 15 15 15 \105 21 35 / and in Maple-input format: -5/3+842/15*b-662/15*b^2+59/15*b^3+(253/105+64/7*b-172/21*b^2+26/35*b^3)*n and in floating point: 3.62035867313195 + 3.26027334785553 n The asymptotic expression for the variance, as a function of n is: 3 2 25936 52546 b 593848 b 794308 b / 416 656 2 92 3 2944\ ----- - -------- + --------- - -------- + |- --- b + --- b - --- b + ----| n 525 1575 1575 1575 \ 105 315 525 1575/ and in Maple-input format: 25936/525-52546/1575*b^3+593848/1575*b^2-794308/1575*b+(-416/105*b+656/315*b^2-\ 92/525*b^3+2944/1575)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 17, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 3, d[1] = 1, d[2] = 3 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = ((h + 7 h + 1) (1 + h ) / ----- n = 0 3 2 4 2 + 4 h (1 + h ) (3 h + 8 h + 3) t 2 2 2 8 6 4 2 2 - (1 + h ) h (4 h + 27 h + 44 h + 27 h + 4) t 2 5 8 6 4 2 3 2 2 4 - 4 (1 + h ) h (10 h + 36 h + 49 h + 36 h + 10) t + 4 (1 + h ) h 2 4 4 3 2 4 3 2 4 (1 + 3 h + h ) (h + h + 3 h + h + 1) (h - h + 3 h - h + 1) t 2 7 12 10 8 6 4 2 5 + 16 (1 + h ) h (2 h + 11 h + 29 h + 41 h + 29 h + 11 h + 2) t - 2 2 8 12 10 8 6 4 2 6 (1 + h ) h (12 h + 76 h + 162 h + 183 h + 162 h + 76 h + 12) t - 2 11 12 10 8 6 4 2 7 4 (1 + h ) h (16 h + 86 h + 176 h + 207 h + 176 h + 86 h + 16) t 2 2 12 12 10 8 6 4 2 8 + (1 + h ) h (8 h + 44 h + 117 h + 169 h + 117 h + 44 h + 8) t + 2 15 12 10 8 6 4 2 9 4 (1 + h ) h (8 h + 48 h + 127 h + 176 h + 127 h + 48 h + 8) t 2 2 18 4 2 4 2 10 - (1 + h ) h (4 h + 9 h + 4) (3 h + 4 h + 3) t 2 21 2 2 4 2 11 - 4 (1 + h ) h (2 + 3 h ) (3 + 2 h ) (2 h + 3 h + 2) t 2 4 24 12 2 3 27 13 2 2 30 14 + 6 (1 + h ) h t + 24 (1 + h ) h t - (1 + h ) h t 33 2 15 / 4 8 2 4 6 2 - 4 h (1 + h ) t ) / ((t h + 1 - 2 t h - 2 h t ) / 4 8 2 4 2 2 (t h - 2 t h - 2 t h + 1) 2 2 4 2 2 6 2 4 8 3 6 (1 + 2 h t - 2 t h - 2 t h - 2 h t + t h + 2 t h ) 2 2 4 2 2 6 2 4 8 3 6 (1 - 2 h t - 2 t h - 2 t h - 2 h t + t h - 2 t h )) and in Maple input format: ((h^4+7*h^2+1)*(1+h^2)^2+4*h^3*(1+h^2)*(3*h^4+8*h^2+3)*t-(1+h^2)^2*h^2*(4*h^8+ 27*h^6+44*h^4+27*h^2+4)*t^2-4*(1+h^2)*h^5*(10*h^8+36*h^6+49*h^4+36*h^2+10)*t^3+ 4*(1+h^2)^2*h^4*(1+3*h^2+h^4)*(h^4+h^3+3*h^2+h+1)*(h^4-h^3+3*h^2-h+1)*t^4+16*(1 +h^2)*h^7*(2*h^12+11*h^10+29*h^8+41*h^6+29*h^4+11*h^2+2)*t^5-(1+h^2)^2*h^8*(12* h^12+76*h^10+162*h^8+183*h^6+162*h^4+76*h^2+12)*t^6-4*(1+h^2)*h^11*(16*h^12+86* h^10+176*h^8+207*h^6+176*h^4+86*h^2+16)*t^7+(1+h^2)^2*h^12*(8*h^12+44*h^10+117* h^8+169*h^6+117*h^4+44*h^2+8)*t^8+4*(1+h^2)*h^15*(8*h^12+48*h^10+127*h^8+176*h^ 6+127*h^4+48*h^2+8)*t^9-(1+h^2)^2*h^18*(4*h^4+9*h^2+4)*(3*h^4+4*h^2+3)*t^10-4*( 1+h^2)*h^21*(2+3*h^2)*(3+2*h^2)*(2*h^4+3*h^2+2)*t^11+6*(1+h^2)^4*h^24*t^12+24*( 1+h^2)^3*h^27*t^13-(1+h^2)^2*h^30*t^14-4*h^33*(1+h^2)*t^15)/(t^4*h^8+1-2*t^2*h^ 4-2*h^6*t^2)/(t^4*h^8-2*t^2*h^4-2*t^2*h^2+1)/(1+2*h^2*t-2*t^2*h^4-2*t^2*h^2-2*h ^6*t^2+t^4*h^8+2*t^3*h^6)/(1-2*h^2*t-2*t^2*h^4-2*t^2*h^2-2*h^6*t^2+t^4*h^8-2*t^ 3*h^6) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 2 - 3 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.267949192 We have the following proven facts The total number of tilings of the region is asymptotic to: / 26 b\ n |97/3 - ----| (1/b) \ 3 / that in floating-point is: n 30.01110700 3.732050814 and in Maple input format: 30.01110700*3.732050814^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 4 + 2 n and in Maple-input format: 4+2*n and in floating point: 4 + 2 n The asymptotic expression for the variance, as a function of n is: 4 b / 4 b\ 17/6 - --- + |8/9 - ---| n 3 \ 9 / and in Maple-input format: 17/6-4/3*b+(8/9-4/9*b)*n and in floating point: 2.476067744 + 0.7698003591 n The even alpha coefficients until the, 8, -th , are: 509 8403 --- - 3 b ---- - 45 b -2 + b 64 -30 + 15 b 64 [1, 3 + ------ + ---------, 15 + ---------- + -----------, n 2 n 2 n n 67557 ----- - 630 b -420 + 210 b 32 105 + ------------ + -------------] n 2 n and in Maple-input format: [1, 3+(-2+b)/n+(509/64-3*b)/n^2, 15+(-30+15*b)/n+(8403/64-45*b)/n^2, 105+(-420+ 210*b)/n+(67557/32-630*b)/n^2] and in floating-point it is: 1.732050808 7.149277424 25.98076212 119.2391614 [1, 3 - ----------- + -----------, 15 - ----------- + -----------, n 2 n 2 n n 363.7306697 1942.348259 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 18, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 3, d[1] = 2, d[2] = 1 Then infinity ----- \ n 2 4 2 ) A(n, h) t = (h (h + 6 h + 4) / ----- n = 0 3 4 2 6 8 - h (9 + 51 h + 38 h + 27 h + 4 h ) t 4 2 6 10 4 8 12 2 + h (48 h + 163 h + 7 + 36 h + 127 h + 110 h + 4 h ) t 5 8 12 2 14 6 4 10 3 - h (250 h + 76 h + 19 h + 12 h + 186 h + 80 h + 192 h + 2) t 8 2 14 6 10 8 12 4 4 + h (13 h + 8 h + 154 h + 1 + 141 h + 201 h + 52 h + 63 h ) t 13 2 8 6 4 2 5 - h (1 + h ) (12 h + 39 h + 43 h + 20 h + 3) t 18 2 2 4 6 23 2 7 / + h (1 + h ) (3 + 10 h + 6 h ) t - h (1 + h ) t ) / ( / 6 2 3 6 2 3 5 4 12 3 11 (h t - 2 h t - h t + 1) (h t + 1 - 2 h t - 2 t h ) (t h - 4 t h 2 10 3 9 2 8 3 7 6 2 5 2 4 + 4 t h - 6 t h + 8 t h - 2 t h + 6 h t - 4 t h + 4 t h 3 2 2 - 6 h t + t h - 2 h t + 1)) and in Maple input format: (h^2*(h^4+6*h^2+4)-h^3*(9+51*h^4+38*h^2+27*h^6+4*h^8)*t+h^4*(48*h^2+163*h^6+7+ 36*h^10+127*h^4+110*h^8+4*h^12)*t^2-h^5*(250*h^8+76*h^12+19*h^2+12*h^14+186*h^6 +80*h^4+192*h^10+2)*t^3+h^8*(13*h^2+8*h^14+154*h^6+1+141*h^10+201*h^8+52*h^12+ 63*h^4)*t^4-h^13*(1+h^2)*(12*h^8+39*h^6+43*h^4+20*h^2+3)*t^5+h^18*(1+h^2)*(3+10 *h^2+6*h^4)*t^6-h^23*(1+h^2)*t^7)/(h^6*t^2-2*h^3*t-h*t+1)/(h^6*t^2+1-2*h^3*t-2* t*h^5)/(t^4*h^12-4*t^3*h^11+4*t^2*h^10-6*t^3*h^9+8*t^2*h^8-2*t^3*h^7+6*h^6*t^2-\ 4*t*h^5+4*t^2*h^4-6*h^3*t+t^2*h^2-2*h*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /253 172 2 26 3\ - 5/3 + ----- - ------ + ----- + |--- + 64/7 b - --- b + -- b | n 15 15 15 \105 21 35 / and in Maple-input format: -5/3+842/15*b-662/15*b^2+59/15*b^3+(253/105+64/7*b-172/21*b^2+26/35*b^3)*n and in floating point: 3.62035867313195 + 3.26027334785553 n The asymptotic expression for the variance, as a function of n is: 3 2 794308 b 25936 52546 b 593848 b /2944 416 656 2 92 3\ - -------- + ----- - -------- + --------- + |---- - --- b + --- b - --- b | n 1575 525 1575 1575 \1575 105 315 525 / and in Maple-input format: -794308/1575*b+25936/525-52546/1575*b^3+593848/1575*b^2+(2944/1575-416/105*b+ 656/315*b^2-92/525*b^3)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 19, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 3, d[1] = 2, d[2] = 2 Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = (-(h + 7 h + 1) (1 + h ) / ----- n = 0 2 12 10 8 6 4 2 + (1 + h ) (4 h + 36 h + 90 h + 109 h + 73 h + 21 h + 2) t + ( 6 16 2 8 14 4 12 18 -345 h - 207 h - 15 h - 774 h - 584 h - 96 h - 1027 h - 44 h 10 20 2 2 18 10 22 2 - 1122 h - 4 h - 1) t + h (526 h + 2255 h + 16 h + 18 h 20 4 8 16 12 14 6 + 136 h + 1 + 134 h + 1381 h + 1228 h + 2504 h + 2021 h + 549 h 3 6 6 14 18 22 10 2 ) t - h (770 h + 2236 h + 583 h + 20 h + 2580 h + 32 h 8 4 16 20 12 4 10 2 + 1724 h + 212 h + 2 + 1348 h + 160 h + 2776 h ) t + h (14 h 16 20 12 14 4 18 10 + 798 h + 68 h + 1634 h + 1397 h + 84 h + 296 h + 1307 h 8 6 22 5 18 + 735 h + 299 h + 8 h + 1) t - h ( 16 14 10 2 12 4 6 8 3 + 12 h + 77 h + 383 h + 32 h + 222 h + 135 h + 305 h + 426 h ) 6 26 2 4 2 4 2 7 34 2 2 8 t + h (1 + h ) (3 h + 5 h + 3) (2 h + 4 h + 1) t - h (1 + h ) t / 4 2 8 4 2 ) / ((t h - 1) (t h - 2 t h - 4 h t - t + 1) / 2 8 4 6 4 16 3 14 3 12 2 12 (t h + 1 - 2 t h - 2 t h ) (t h - 4 t h - 12 t h + 4 t h 3 10 2 10 3 8 2 8 6 2 6 2 4 - 10 t h + 8 t h - 2 t h + 6 t h + 16 h t - 4 t h + 20 t h 4 2 2 2 2 - 12 t h + 8 t h - 10 h t + t - 2 t + 1)) and in Maple input format: (-(h^4+7*h^2+1)*(1+h^2)^2+(1+h^2)*(4*h^12+36*h^10+90*h^8+109*h^6+73*h^4+21*h^2+ 2)*t+(-345*h^6-207*h^16-15*h^2-774*h^8-584*h^14-96*h^4-1027*h^12-44*h^18-1122*h ^10-4*h^20-1)*t^2+h^2*(526*h^18+2255*h^10+16*h^22+18*h^2+136*h^20+1+134*h^4+ 1381*h^8+1228*h^16+2504*h^12+2021*h^14+549*h^6)*t^3-h^6*(770*h^6+2236*h^14+583* h^18+20*h^22+2580*h^10+32*h^2+1724*h^8+212*h^4+2+1348*h^16+160*h^20+2776*h^12)* t^4+h^10*(14*h^2+798*h^16+68*h^20+1634*h^12+1397*h^14+84*h^4+296*h^18+1307*h^10 +735*h^8+299*h^6+8*h^22+1)*t^5-h^18*(3+12*h^16+77*h^14+383*h^10+32*h^2+222*h^12 +135*h^4+305*h^6+426*h^8)*t^6+h^26*(1+h^2)*(3*h^4+5*h^2+3)*(2*h^4+4*h^2+1)*t^7- h^34*(1+h^2)^2*t^8)/(t*h^4-1)/(t^2*h^8-2*t*h^4-4*h^2*t-t+1)/(t^2*h^8+1-2*t*h^4-\ 2*t*h^6)/(t^4*h^16-4*t^3*h^14-12*t^3*h^12+4*t^2*h^12-10*t^3*h^10+8*t^2*h^10-2*t ^3*h^8+6*t^2*h^8+16*h^6*t^2-4*t*h^6+20*t^2*h^4-12*t*h^4+8*t^2*h^2-10*h^2*t+t^2-\ 2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 28 _Z + 63 _Z - 28 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 28 b + 63 b - 28 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0390932603206857 We have the following proven facts The total number of tilings of the region is asymptotic to: /63647 17041 7687 2 1237 3\ n |----- - ----- b + ---- b - ---- b | (1/b) \1980 220 220 990 / that in floating-point is: n 29.1701460305608 25.5798567783016 and in Maple input format: HFloat(29.1701460305607725)*HFloat(25.5798567783015649)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 2239 8996 b 9484 b 119 b /604 256 184 2 34 3\ ---- - ------ + ------- - ------ + |--- - --- b + --- b - --- b | n 495 495 495 165 \165 33 33 165 / and in Maple-input format: 2239/495-8996/495*b+9484/495*b^2-119/165*b^3+(604/165-256/33*b+184/33*b^2-34/ 165*b^3)*n and in floating point: 3.84199987246269 + 3.36584615677336 n The asymptotic expression for the variance, as a function of n is: 3 2 1456 11698 b 917648 b 509272 b ---- + -------- - --------- + -------- 7425 2475 7425 7425 / 1856 608 2 316 3 17416\ + |- ---- b + --- b - ---- b + -----| n \ 495 495 7425 7425 / and in Maple-input format: 1456/7425+11698/2475*b^3-917648/7425*b^2+509272/7425*b+(-1856/495*b+608/495*b^2 -316/7425*b^3+17416/7425)*n and in floating point: 2.68885841619758 + 2.20088386493014 n The even alpha coefficients until the, 8, -th , are: 1434034 338320 44840 2 6641 3 - ------- + ------- b + ------- b - ------- b 3110217 1036739 1036739 3110217 / 21969685269856 [1, 3 + ----------------------------------------------- + |- -------------- n \ 4774183095 120652827161038 211191088530932 2 5381346012815 3\ / 2 + --------------- b - --------------- b + ------------- b | / n , 15 954836619 954836619 636557746 / / 230378218 1170459120 676704360 2 24690643 3 - --------- + ---------- b - --------- b + -------- b 28934443 28934443 28934443 28934443 / + ------------------------------------------------------- + | n \ 6957636763169604 191048766088465162 334457385536666108 2 - ---------------- + ------------------ b - ------------------ b 97711614011 97711614011 97711614011 25566937383188123 3\ / 2 + ----------------- b | / n , 105 195423228022 / / 40139100484 338689918560 211315777680 2 7747433134 3 - ----------- + ------------ b - ------------ b + ---------- b 318278873 318278873 318278873 318278873 / + ---------------------------------------------------------------- + | n \ 100381232738332861 2756365931861602956 4826233409563180104 2 - ------------------ + ------------------- b - ------------------- b 97711614011 97711614011 97711614011 184466688465337117 3\ / 2 + ------------------ b | / n ] 97711614011 / / and in Maple-input format: [1, 3+(-1434034/3110217+338320/1036739*b+44840/1036739*b^2-6641/3110217*b^3)/n+ (-21969685269856/4774183095+120652827161038/954836619*b-211191088530932/ 954836619*b^2+5381346012815/636557746*b^3)/n^2, 15+(-230378218/28934443+ 1170459120/28934443*b-676704360/28934443*b^2+24690643/28934443*b^3)/n+(-\ 6957636763169604/97711614011+191048766088465162/97711614011*b-\ 334457385536666108/97711614011*b^2+25566937383188123/195423228022*b^3)/n^2, 105 +(-40139100484/318278873+338689918560/318278873*b-211315777680/318278873*b^2+ 7747433134/318278873*b^3)/n+(-100381232738332861/97711614011+ 2756365931861602956/97711614011*b-4826233409563180104/97711614011*b^2+ 184466688465337117/97711614011*b^3)/n^2] and in floating-point it is: 0.448248696679244 0.521390865623671 [1, 3 - ----------------- + -----------------, n 2 n 6.41636251823189 7.17009859652749 15 - ---------------- + ----------------, n 2 n 85.5259242078518 94.8577919234394 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 166459104 1131708672 765023616 2 140895504 3 - ---------- + ---------- b - --------- b + ---------- b 1591394365 318278873 318278873 1591394365 / 9690578062688 [---------------------------------------------------------- + |- ------------- n \ 44414370005 265995727357136 467337563593024 2 17863811162032 3\ / 2 + --------------- b - --------------- b + -------------- b | / n , 44414370005 44414370005 44414370005 / / 3329182080 113170867200 76502361600 2 2817910080 3 - ---------- + ------------ b - ----------- b + ---------- b 318278873 318278873 318278873 318278873 / -------------------------------------------------------------- + | n \ 193611642474400 5314424312751520 9344271811167680 2 - --------------- + ---------------- b - ---------------- b 8882874001 8882874001 8882874001 357187671073040 3\ / 2 + --------------- b | / n , 8882874001 / / 367042324320 12477088108800 8434385366400 2 310674586320 3 - ------------ + -------------- b - ------------- b + ------------ b 318278873 318278873 318278873 318278873 / ---------------------------------------------------------------------- + | n \ 6546032251382902320 179674484780934480960 316182229067425724640 2 - ------------------- + --------------------- b - --------------------- b 2727042318307 2727042318307 2727042318307 12086391703848061320 3\ / 2 + -------------------- b | / n ] 2727042318307 / / and in Maple-input format: [(-166459104/1591394365+1131708672/318278873*b-765023616/318278873*b^2+ 140895504/1591394365*b^3)/n+(-9690578062688/44414370005+265995727357136/ 44414370005*b-467337563593024/44414370005*b^2+17863811162032/44414370005*b^3)/n ^2, (-3329182080/318278873+113170867200/318278873*b-76502361600/318278873*b^2+ 2817910080/318278873*b^3)/n+(-193611642474400/8882874001+5314424312751520/ 8882874001*b-9344271811167680/8882874001*b^2+357187671073040/8882874001*b^3)/n^ 2, (-367042324320/318278873+12477088108800/318278873*b-8434385366400/318278873* b^2+310674586320/318278873*b^3)/n+(-6546032251382902320/2727042318307+ 179674484780934480960/2727042318307*b-316182229067425724640/2727042318307*b^2+ 12086391703848061320/2727042318307*b^3)/n^2] and in floating-point it is: 0.0307367931956769 0.114706617065132 3.07367931956768 12.7009262232392 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 338.873145154837 1637.18096304436 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 20, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 3, d[1] = 2, d[2] = 3 Then infinity ----- \ n 2 2 4 4 2 3 ) A(n, h) t = (-h (1 + 3 h + h ) (h + 9 h + 9) + h / ----- n = 0 8 2 4 12 14 6 10 (1369 h + 290 h + 901 h + 160 h + 12 h + 38 + 1482 h + 683 h ) t 4 2 18 16 14 12 10 - h (1 + h ) (60 h + 812 h + 3838 h + 9348 h + 13569 h 8 6 4 2 2 5 12 + 12448 h + 7300 h + 2669 h + 560 h + 52) t + h (214903 h 24 26 8 6 2 10 + 2520 h + 160 h + 55553 h + 16722 h + 24 + 416 h + 129240 h 22 16 14 20 4 18 + 16028 h + 222031 h + 257445 h + 57982 h + 3376 h + 136379 h ) 3 8 18 16 30 10 24 t - h (1358216 h + 1582719 h + 240 h + 469510 h + 138832 h 28 4 6 12 2 14 + 4160 h + 7476 h + 44413 h + 934994 h + 732 h + 1397020 h 20 8 26 22 4 13 2 + 873735 h + 172514 h + 32 + 31224 h + 413108 h ) t + h (1 + h ) 30 28 26 24 22 20 (192 h + 3712 h + 31824 h + 161632 h + 546780 h + 1314526 h 18 16 14 12 10 + 2340365 h + 3167760 h + 3302566 h + 2650771 h + 1615142 h 8 6 4 2 5 18 2 + 727295 h + 232583 h + 49794 h + 6430 h + 384) t - h (1 + h ) ( 32 30 28 26 24 22 64 h + 1856 h + 21280 h + 136992 h + 575924 h + 1718192 h 20 18 16 14 12 + 3828954 h + 6582857 h + 8886233 h + 9460802 h + 7889083 h 10 8 6 4 2 6 + 5068414 h + 2445678 h + 853612 h + 202930 h + 29369 h + 1952) t 23 22 20 24 8 + h (5496 + 1018536 h + 2612628 h + 292784 h + 3876692 h 2 18 28 4 12 26 + 72581 h + 5153346 h + 6976 h + 434542 h + 9353738 h + 57888 h 10 6 14 16 30 + 6932564 h + 1573249 h + 9756369 h + 7982091 h + 384 h ) 2 2 7 28 2 32 30 28 26 (1 + h ) t - h (1 + h ) (896 h + 14848 h + 115488 h + 567792 h 24 22 20 18 16 + 1996288 h + 5343380 h + 11243718 h + 18889635 h + 25486087 h 14 12 10 8 6 + 27586166 h + 23769980 h + 16054287 h + 8284514 h + 3137134 h 4 2 8 33 2 32 30 + 816518 h + 129667 h + 9414) t + h (1 + h ) (1024 h + 16320 h 28 26 24 22 20 + 123936 h + 601008 h + 2096240 h + 5584692 h + 11721814 h 18 16 14 12 10 + 19670011 h + 26529668 h + 28725766 h + 24784507 h + 16782126 h 8 6 4 2 9 38 + 8693377 h + 3308791 h + 866686 h + 138677 h + 10154) t - h 2 32 30 28 26 24 (1 + h ) (576 h + 9856 h + 80928 h + 418768 h + 1524820 h 22 20 18 16 14 + 4156940 h + 8814306 h + 14847323 h + 20029533 h + 21611586 h 12 10 8 6 4 + 18497082 h + 12370246 h + 6309404 h + 2361200 h + 608090 h 2 10 43 2 32 30 28 + 95724 h + 6900) t + h (1 + h ) (128 h + 3136 h + 31776 h 26 24 22 20 18 + 188432 h + 752120 h + 2174140 h + 4758518 h + 8110263 h 16 14 12 10 8 + 10926750 h + 11671801 h + 9812498 h + 6390401 h + 3146061 h 6 4 2 11 48 10 + 1128145 h + 277171 h + 41568 h + 2856) t - h (3327213 h 2 22 18 28 20 + 11296 h + 2590180 h + 7446592 h + 59472 h + 4983921 h 24 8 16 26 30 + 1022056 h + 1406554 h + 8753727 h + 295896 h + 7472 h 6 4 32 12 14 12 + 429953 h + 656 + 89384 h + 448 h + 5917899 h + 8125516 h ) t + 53 14 28 2 24 22 6 h (2132859 h + 9056 h + 1456 h + 234308 h + 655838 h + 79990 h 12 4 20 18 30 + 1464932 h + 14224 h + 64 + 1339069 h + 2048303 h + 672 h 26 10 16 8 13 60 2 + 58160 h + 763966 h + 2385202 h + 294390 h ) t - h (1 + h ) ( 24 22 20 18 16 14 560 h + 5820 h + 27896 h + 81510 h + 160623 h + 222607 h 12 10 8 6 4 2 + 220448 h + 156619 h + 79599 h + 28494 h + 6860 h + 992 h + 64) 14 67 10 2 12 16 20 t + h (41997 h + 16 + 264 h + 53361 h + 28581 h + 2668 h 6 22 18 14 8 4 15 + 8352 h + 280 h + 11358 h + 47163 h + 22835 h + 1952 h ) t - 76 4 14 10 16 12 6 h (1068 h + 637 h + 3633 h + 84 h + 24 + 2043 h + 2621 h 2 8 16 85 2 2 2 3 17 + 244 h + 3917 h ) t + 2 h (2 + h ) (7 h + 3) (1 + h ) t 94 2 2 18 / 5 - h (2 + h ) (1 + h ) t ) / ((t h - 1) / 2 10 5 7 4 20 3 15 3 13 3 11 (t h + 1 - 2 t h - 2 t h ) (t h - 4 t h - 6 t h - 2 t h 2 10 2 8 6 2 5 3 + 6 t h + 12 t h + 5 h t - 4 t h - 6 h t - 2 h t + 1) (1 3 13 3 15 4 20 2 8 3 6 2 5 - 4 t h - 8 t h + t h + 8 t h - 4 h t + 4 h t - 8 t h 2 10 3 17 7 2 14 2 12 3 13 + 6 t h - 4 t h - 4 t h + 4 t h + 8 t h ) (1 - 424 t h 3 11 3 15 4 20 2 8 4 24 2 4 - 320 t h - 380 t h + 662 t h + 112 t h + 144 t h + 24 t h 2 2 3 6 2 7 31 5 2 10 + 4 t h - 16 h t + 62 h t - 4 t h - 20 t h + 144 t h 4 12 4 26 7 35 7 33 6 32 + 33 t h - 4 h t + 64 t h - 20 t h - 16 t h + 96 t h 3 9 3 7 8 40 6 34 6 26 6 24 - 128 t h - 20 t h + t h + 24 t h + 62 t h + 24 t h 3 17 6 22 6 30 3 19 3 21 7 - 280 t h + 4 t h + 144 t h - 144 t h - 32 t h - 8 t h 6 28 5 23 5 25 2 14 5 17 2 12 + 112 t h - 424 t h - 380 t h + 24 t h - 20 t h + 96 t h 4 16 5 19 5 31 5 29 5 27 + 516 t h - 128 t h - 32 t h - 144 t h - 280 t h 5 21 7 37 4 22 4 14 4 28 4 18 - 320 t h - 8 t h + 352 t h + 200 t h + 16 t h + 752 t h )) and in Maple input format: (-h^2*(1+3*h^2+h^4)*(h^4+9*h^2+9)+h^3*(1369*h^8+290*h^2+901*h^4+160*h^12+12*h^ 14+38+1482*h^6+683*h^10)*t-h^4*(1+h^2)*(60*h^18+812*h^16+3838*h^14+9348*h^12+ 13569*h^10+12448*h^8+7300*h^6+2669*h^4+560*h^2+52)*t^2+h^5*(214903*h^12+2520*h^ 24+160*h^26+55553*h^8+16722*h^6+24+416*h^2+129240*h^10+16028*h^22+222031*h^16+ 257445*h^14+57982*h^20+3376*h^4+136379*h^18)*t^3-h^8*(1358216*h^18+1582719*h^16 +240*h^30+469510*h^10+138832*h^24+4160*h^28+7476*h^4+44413*h^6+934994*h^12+732* h^2+1397020*h^14+873735*h^20+172514*h^8+32+31224*h^26+413108*h^22)*t^4+h^13*(1+ h^2)*(192*h^30+3712*h^28+31824*h^26+161632*h^24+546780*h^22+1314526*h^20+ 2340365*h^18+3167760*h^16+3302566*h^14+2650771*h^12+1615142*h^10+727295*h^8+ 232583*h^6+49794*h^4+6430*h^2+384)*t^5-h^18*(1+h^2)*(64*h^32+1856*h^30+21280*h^ 28+136992*h^26+575924*h^24+1718192*h^22+3828954*h^20+6582857*h^18+8886233*h^16+ 9460802*h^14+7889083*h^12+5068414*h^10+2445678*h^8+853612*h^6+202930*h^4+29369* h^2+1952)*t^6+h^23*(5496+1018536*h^22+2612628*h^20+292784*h^24+3876692*h^8+ 72581*h^2+5153346*h^18+6976*h^28+434542*h^4+9353738*h^12+57888*h^26+6932564*h^ 10+1573249*h^6+9756369*h^14+7982091*h^16+384*h^30)*(1+h^2)^2*t^7-h^28*(1+h^2)*( 896*h^32+14848*h^30+115488*h^28+567792*h^26+1996288*h^24+5343380*h^22+11243718* h^20+18889635*h^18+25486087*h^16+27586166*h^14+23769980*h^12+16054287*h^10+ 8284514*h^8+3137134*h^6+816518*h^4+129667*h^2+9414)*t^8+h^33*(1+h^2)*(1024*h^32 +16320*h^30+123936*h^28+601008*h^26+2096240*h^24+5584692*h^22+11721814*h^20+ 19670011*h^18+26529668*h^16+28725766*h^14+24784507*h^12+16782126*h^10+8693377*h ^8+3308791*h^6+866686*h^4+138677*h^2+10154)*t^9-h^38*(1+h^2)*(576*h^32+9856*h^ 30+80928*h^28+418768*h^26+1524820*h^24+4156940*h^22+8814306*h^20+14847323*h^18+ 20029533*h^16+21611586*h^14+18497082*h^12+12370246*h^10+6309404*h^8+2361200*h^6 +608090*h^4+95724*h^2+6900)*t^10+h^43*(1+h^2)*(128*h^32+3136*h^30+31776*h^28+ 188432*h^26+752120*h^24+2174140*h^22+4758518*h^20+8110263*h^18+10926750*h^16+ 11671801*h^14+9812498*h^12+6390401*h^10+3146061*h^8+1128145*h^6+277171*h^4+ 41568*h^2+2856)*t^11-h^48*(3327213*h^10+11296*h^2+2590180*h^22+7446592*h^18+ 59472*h^28+4983921*h^20+1022056*h^24+1406554*h^8+8753727*h^16+295896*h^26+7472* h^30+429953*h^6+656+89384*h^4+448*h^32+5917899*h^12+8125516*h^14)*t^12+h^53*( 2132859*h^14+9056*h^28+1456*h^2+234308*h^24+655838*h^22+79990*h^6+1464932*h^12+ 14224*h^4+64+1339069*h^20+2048303*h^18+672*h^30+58160*h^26+763966*h^10+2385202* h^16+294390*h^8)*t^13-h^60*(1+h^2)*(560*h^24+5820*h^22+27896*h^20+81510*h^18+ 160623*h^16+222607*h^14+220448*h^12+156619*h^10+79599*h^8+28494*h^6+6860*h^4+ 992*h^2+64)*t^14+h^67*(41997*h^10+16+264*h^2+53361*h^12+28581*h^16+2668*h^20+ 8352*h^6+280*h^22+11358*h^18+47163*h^14+22835*h^8+1952*h^4)*t^15-h^76*(1068*h^4 +637*h^14+3633*h^10+84*h^16+24+2043*h^12+2621*h^6+244*h^2+3917*h^8)*t^16+2*h^85 *(2+h^2)*(7*h^2+3)*(1+h^2)^3*t^17-h^94*(2+h^2)*(1+h^2)*t^18)/(t*h^5-1)/(t^2*h^ 10+1-2*t*h^5-2*t*h^7)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-2*t^3*h^11+6*t^2*h^10+12* t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1)/(1-4*t^3*h^13-8*t^3*h^15+t^4*h^20+8* t^2*h^8-4*h^3*t+4*h^6*t^2-8*t*h^5+6*t^2*h^10-4*t^3*h^17-4*t*h^7+4*t^2*h^14+8*t^ 2*h^12)/(1-424*t^3*h^13-320*t^3*h^11-380*t^3*h^15+662*t^4*h^20+112*t^2*h^8+144* t^4*h^24+24*t^2*h^4+4*t^2*h^2-16*h^3*t+62*h^6*t^2-4*t^7*h^31-20*t*h^5+144*t^2*h ^10+33*t^4*h^12-4*h*t+64*t^4*h^26-20*t^7*h^35-16*t^7*h^33+96*t^6*h^32-128*t^3*h ^9-20*t^3*h^7+t^8*h^40+24*t^6*h^34+62*t^6*h^26+24*t^6*h^24-280*t^3*h^17+4*t^6*h ^22+144*t^6*h^30-144*t^3*h^19-32*t^3*h^21-8*t*h^7+112*t^6*h^28-424*t^5*h^23-380 *t^5*h^25+24*t^2*h^14-20*t^5*h^17+96*t^2*h^12+516*t^4*h^16-128*t^5*h^19-32*t^5* h^31-144*t^5*h^29-280*t^5*h^27-320*t^5*h^21-8*t^7*h^37+352*t^4*h^22+200*t^4*h^ 14+16*t^4*h^28+752*t^4*h^18) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /1764901 1624521 1013333 2 3219 3\ n |------- - ------- b + ------- b - ---- b | (1/b) \ 22440 3740 11220 1496 / that in floating-point is: n 66.8056926239638 36.4645094590774 and in Maple input format: HFloat(66.8056926239637647)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 15583 502897 b 107777 b 1289 b /3541 2184 468 2 56 3\ ----- + -------- - --------- + ------- + |---- + ---- b - --- b + --- b | n 5610 5610 5610 2805 \935 187 187 935 / and in Maple-input format: 15583/5610+502897/5610*b-107777/5610*b^2+1289/2805*b^3+(3541/935+2184/187*b-468 /187*b^2+56/935*b^3)*n and in floating point: 5.22164119915843 + 4.10557280859901 n The asymptotic expression for the variance, as a function of n is: 2 3 36273007 b 7765187 b 222736 185698 b - ---------- + ---------- + ------ - --------- 42075 42075 8415 42075 /105574 26128 15728 2 622 3\ + |------ - ----- b + ----- b - ----- b | n \42075 2805 8415 14025 / and in Maple-input format: -36273007/42075*b+7765187/42075*b^2+222736/8415-185698/42075*b^3+(105574/42075-\ 26128/2805*b+15728/8415*b^2-622/14025*b^3)*n and in floating point: 2.96536890504092 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 /755497810818251 [1, 3 + ----------------------------------------------- + |--------------- n \ 28160464640 395868535803137 593747659641319 2 28400798598787 3\ / 2 - --------------- b + --------------- b - -------------- b | / n , 15 402292352 2816046464 5632092928 / / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b 25143272 22000363 22000363 176002904 / + ------------------------------------------------------- + | n \ 109528284629096541 286954640515303311 430392155354112057 2 ------------------ - ------------------ b + ------------------ b 275972553472 19712325248 137986276736 20586997557395961 3\ / 2 - ----------------- b | / n , 105 275972553472 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 107998196881527861 1980614777716651269 424378414252057029 2 ------------------ - ------------------- b + ------------------ b 19712325248 9856162624 9856162624 20299343067550257 3\ / 2 - ----------------- b | / n ] 19712325248 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(755497810818251/28160464640-395868535803137/402292352*b+593747659641319/ 2816046464*b^2-28400798598787/5632092928*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 109528284629096541/275972553472-286954640515303311/19712325248*b+ 430392155354112057/137986276736*b^2-20586997557395961/275972553472*b^3)/n^2, 105+(-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752* b^3)/n+(107998196881527861/19712325248-1980614777716651269/9856162624*b+ 424378414252057029/9856162624*b^2-20299343067550257/19712325248*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.761238608831915 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 12.0047124948301 15 - ---------------- + ----------------, n 2 n 106.402545323854 184.604855321300 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 47788542835569 [------------------------------------------------------ + |- -------------- n \ 86241422960 1001579894699661 1502215360621407 2 17963882819019 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 49280813120 344965691840 172482845920 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 119464631934525 5007585548893905 7510619902940235 2 - --------------- + ---------------- b - ---------------- b 2156035574 2464040656 17248284592 89813974807395 3\ / 2 + -------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 210710464120926135 15456404634254124435 473109292840243905 2 - ------------------ + -------------------- b - ------------------ b 34496569184 68993138368 9856162624 9900745700744655 3\ / 2 + ---------------- b | / n ] 8624142296 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-47788542835569/86241422960+1001579894699661/49280813120*b-\ 1502215360621407/344965691840*b^2+17963882819019/172482845920*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-119464631934525/2156035574+5007585548893905/2464040656*b-\ 7510619902940235/17248284592*b^2+89813974807395/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-210710464120926135/34496569184+15456404634254124435/ 68993138368*b-473109292840243905/9856162624*b^2+9900745700744655/8624142296*b^3 )/n^2] and in floating-point it is: 0.00714279180461598 0.0360480831957141 0.714279180461598 3.95954221793470 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 502.750599341107 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 21, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 3, d[1] = 3, d[2] = 1 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = ((h + 7 h + 1) (1 + h ) / ----- n = 0 3 2 4 2 + 4 h (1 + h ) (3 h + 8 h + 3) t 2 2 2 8 6 4 2 2 - (1 + h ) h (4 h + 27 h + 44 h + 27 h + 4) t 2 5 8 6 4 2 3 2 2 4 - 4 (1 + h ) h (10 h + 36 h + 49 h + 36 h + 10) t + 4 (1 + h ) h 2 4 4 3 2 4 3 2 4 (1 + 3 h + h ) (h + h + 3 h + h + 1) (h - h + 3 h - h + 1) t 2 7 12 10 8 6 4 2 5 + 16 (1 + h ) h (2 h + 11 h + 29 h + 41 h + 29 h + 11 h + 2) t - 2 2 8 12 10 8 6 4 2 6 (1 + h ) h (12 h + 76 h + 162 h + 183 h + 162 h + 76 h + 12) t - 2 11 12 10 8 6 4 2 7 4 (1 + h ) h (16 h + 86 h + 176 h + 207 h + 176 h + 86 h + 16) t 2 2 12 12 10 8 6 4 2 8 + (1 + h ) h (8 h + 44 h + 117 h + 169 h + 117 h + 44 h + 8) t + 2 15 12 10 8 6 4 2 9 4 (1 + h ) h (8 h + 48 h + 127 h + 176 h + 127 h + 48 h + 8) t 2 2 18 4 2 4 2 10 - (1 + h ) h (4 h + 9 h + 4) (3 h + 4 h + 3) t 2 21 2 2 4 2 11 - 4 (1 + h ) h (2 + 3 h ) (3 + 2 h ) (2 h + 3 h + 2) t 2 4 24 12 2 3 27 13 2 2 30 14 + 6 (1 + h ) h t + 24 (1 + h ) h t - (1 + h ) h t 33 2 15 / 4 8 2 4 6 2 - 4 h (1 + h ) t ) / ((t h + 1 - 2 t h - 2 h t ) / 4 8 2 4 2 2 (t h - 2 t h - 2 t h + 1) 2 2 4 2 2 6 2 4 8 3 6 (1 + 2 h t - 2 t h - 2 t h - 2 h t + t h + 2 t h ) 2 2 4 2 2 6 2 4 8 3 6 (1 - 2 h t - 2 t h - 2 t h - 2 h t + t h - 2 t h )) and in Maple input format: ((h^4+7*h^2+1)*(1+h^2)^2+4*h^3*(1+h^2)*(3*h^4+8*h^2+3)*t-(1+h^2)^2*h^2*(4*h^8+ 27*h^6+44*h^4+27*h^2+4)*t^2-4*(1+h^2)*h^5*(10*h^8+36*h^6+49*h^4+36*h^2+10)*t^3+ 4*(1+h^2)^2*h^4*(1+3*h^2+h^4)*(h^4+h^3+3*h^2+h+1)*(h^4-h^3+3*h^2-h+1)*t^4+16*(1 +h^2)*h^7*(2*h^12+11*h^10+29*h^8+41*h^6+29*h^4+11*h^2+2)*t^5-(1+h^2)^2*h^8*(12* h^12+76*h^10+162*h^8+183*h^6+162*h^4+76*h^2+12)*t^6-4*(1+h^2)*h^11*(16*h^12+86* h^10+176*h^8+207*h^6+176*h^4+86*h^2+16)*t^7+(1+h^2)^2*h^12*(8*h^12+44*h^10+117* h^8+169*h^6+117*h^4+44*h^2+8)*t^8+4*(1+h^2)*h^15*(8*h^12+48*h^10+127*h^8+176*h^ 6+127*h^4+48*h^2+8)*t^9-(1+h^2)^2*h^18*(4*h^4+9*h^2+4)*(3*h^4+4*h^2+3)*t^10-4*( 1+h^2)*h^21*(2+3*h^2)*(3+2*h^2)*(2*h^4+3*h^2+2)*t^11+6*(1+h^2)^4*h^24*t^12+24*( 1+h^2)^3*h^27*t^13-(1+h^2)^2*h^30*t^14-4*h^33*(1+h^2)*t^15)/(t^4*h^8+1-2*t^2*h^ 4-2*h^6*t^2)/(t^4*h^8-2*t^2*h^4-2*t^2*h^2+1)/(1+2*h^2*t-2*t^2*h^4-2*t^2*h^2-2*h ^6*t^2+t^4*h^8+2*t^3*h^6)/(1-2*h^2*t-2*t^2*h^4-2*t^2*h^2-2*h^6*t^2+t^4*h^8-2*t^ 3*h^6) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 2 - 3 2 whose minimal polynomial is, b - 4 b + 1 and whose floating-point appx. , to, 10, digits is, 0.267949192 We have the following proven facts The total number of tilings of the region is asymptotic to: / 26 b\ n |97/3 - ----| (1/b) \ 3 / that in floating-point is: n 30.01110700 3.732050814 and in Maple input format: 30.01110700*3.732050814^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 4 + 2 n and in Maple-input format: 4+2*n and in floating point: 4 + 2 n The asymptotic expression for the variance, as a function of n is: 4 b / 4 b\ 17/6 - --- + |8/9 - ---| n 3 \ 9 / and in Maple-input format: 17/6-4/3*b+(8/9-4/9*b)*n and in floating point: 2.476067744 + 0.7698003591 n The even alpha coefficients until the, 8, -th , are: 509 8403 --- - 3 b ---- - 45 b -2 + b 64 -30 + 15 b 64 [1, 3 + ------ + ---------, 15 + ---------- + -----------, n 2 n 2 n n 67557 ----- - 630 b -420 + 210 b 32 105 + ------------ + -------------] n 2 n and in Maple-input format: [1, 3+(-2+b)/n+(509/64-3*b)/n^2, 15+(-30+15*b)/n+(8403/64-45*b)/n^2, 105+(-420+ 210*b)/n+(67557/32-630*b)/n^2] and in floating-point it is: 1.732050808 7.149277424 25.98076212 119.2391614 [1, 3 - ----------- + -----------, 15 - ----------- + -----------, n 2 n 2 n n 363.7306697 1942.348259 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 22, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 1, c[2] = 3, d[1] = 3, d[2] = 2 Then infinity ----- \ n 2 2 4 4 2 3 ) A(n, h) t = (-h (1 + 3 h + h ) (h + 9 h + 9) + h / ----- n = 0 8 2 4 12 14 6 10 (1369 h + 290 h + 901 h + 160 h + 12 h + 38 + 1482 h + 683 h ) t 4 2 18 16 14 12 10 - h (1 + h ) (60 h + 812 h + 3838 h + 9348 h + 13569 h 8 6 4 2 2 5 12 + 12448 h + 7300 h + 2669 h + 560 h + 52) t + h (214903 h 24 26 8 6 2 10 + 2520 h + 160 h + 55553 h + 16722 h + 24 + 416 h + 129240 h 22 16 14 20 4 18 + 16028 h + 222031 h + 257445 h + 57982 h + 3376 h + 136379 h ) 3 8 18 16 30 10 24 t - h (1358216 h + 1582719 h + 240 h + 469510 h + 138832 h 28 4 6 12 2 14 + 4160 h + 7476 h + 44413 h + 934994 h + 732 h + 1397020 h 20 8 26 22 4 13 2 + 873735 h + 172514 h + 32 + 31224 h + 413108 h ) t + h (1 + h ) 30 28 26 24 22 20 (192 h + 3712 h + 31824 h + 161632 h + 546780 h + 1314526 h 18 16 14 12 10 + 2340365 h + 3167760 h + 3302566 h + 2650771 h + 1615142 h 8 6 4 2 5 18 2 + 727295 h + 232583 h + 49794 h + 6430 h + 384) t - h (1 + h ) ( 32 30 28 26 24 22 64 h + 1856 h + 21280 h + 136992 h + 575924 h + 1718192 h 20 18 16 14 12 + 3828954 h + 6582857 h + 8886233 h + 9460802 h + 7889083 h 10 8 6 4 2 6 + 5068414 h + 2445678 h + 853612 h + 202930 h + 29369 h + 1952) t 23 22 20 24 8 + h (5496 + 1018536 h + 2612628 h + 292784 h + 3876692 h 2 18 28 4 12 26 + 72581 h + 5153346 h + 6976 h + 434542 h + 9353738 h + 57888 h 10 6 14 16 30 + 6932564 h + 1573249 h + 9756369 h + 7982091 h + 384 h ) 2 2 7 28 2 32 30 28 26 (1 + h ) t - h (1 + h ) (896 h + 14848 h + 115488 h + 567792 h 24 22 20 18 16 + 1996288 h + 5343380 h + 11243718 h + 18889635 h + 25486087 h 14 12 10 8 6 + 27586166 h + 23769980 h + 16054287 h + 8284514 h + 3137134 h 4 2 8 33 2 32 30 + 816518 h + 129667 h + 9414) t + h (1 + h ) (1024 h + 16320 h 28 26 24 22 20 + 123936 h + 601008 h + 2096240 h + 5584692 h + 11721814 h 18 16 14 12 10 + 19670011 h + 26529668 h + 28725766 h + 24784507 h + 16782126 h 8 6 4 2 9 38 + 8693377 h + 3308791 h + 866686 h + 138677 h + 10154) t - h 2 32 30 28 26 24 (1 + h ) (576 h + 9856 h + 80928 h + 418768 h + 1524820 h 22 20 18 16 14 + 4156940 h + 8814306 h + 14847323 h + 20029533 h + 21611586 h 12 10 8 6 4 + 18497082 h + 12370246 h + 6309404 h + 2361200 h + 608090 h 2 10 43 2 32 30 28 + 95724 h + 6900) t + h (1 + h ) (128 h + 3136 h + 31776 h 26 24 22 20 18 + 188432 h + 752120 h + 2174140 h + 4758518 h + 8110263 h 16 14 12 10 8 + 10926750 h + 11671801 h + 9812498 h + 6390401 h + 3146061 h 6 4 2 11 48 10 + 1128145 h + 277171 h + 41568 h + 2856) t - h (3327213 h 2 22 18 28 20 + 11296 h + 2590180 h + 7446592 h + 59472 h + 4983921 h 24 8 16 26 30 + 1022056 h + 1406554 h + 8753727 h + 295896 h + 7472 h 6 4 32 12 14 12 + 429953 h + 656 + 89384 h + 448 h + 5917899 h + 8125516 h ) t + 53 14 28 2 24 22 6 h (2132859 h + 9056 h + 1456 h + 234308 h + 655838 h + 79990 h 12 4 20 18 30 + 1464932 h + 14224 h + 64 + 1339069 h + 2048303 h + 672 h 26 10 16 8 13 60 2 + 58160 h + 763966 h + 2385202 h + 294390 h ) t - h (1 + h ) ( 24 22 20 18 16 14 560 h + 5820 h + 27896 h + 81510 h + 160623 h + 222607 h 12 10 8 6 4 2 + 220448 h + 156619 h + 79599 h + 28494 h + 6860 h + 992 h + 64) 14 67 10 2 12 16 20 t + h (41997 h + 16 + 264 h + 53361 h + 28581 h + 2668 h 6 22 18 14 8 4 15 + 8352 h + 280 h + 11358 h + 47163 h + 22835 h + 1952 h ) t - 76 4 14 10 16 12 6 h (1068 h + 637 h + 3633 h + 84 h + 24 + 2043 h + 2621 h 2 8 16 85 2 2 2 3 17 + 244 h + 3917 h ) t + 2 h (2 + h ) (7 h + 3) (1 + h ) t 94 2 2 18 / 5 - h (2 + h ) (1 + h ) t ) / ((t h - 1) / 2 10 5 7 4 20 3 15 3 13 3 11 (t h + 1 - 2 t h - 2 t h ) (t h - 4 t h - 6 t h - 2 t h 2 10 2 8 6 2 5 3 + 6 t h + 12 t h + 5 h t - 4 t h - 6 h t - 2 h t + 1) (1 3 13 3 15 4 20 2 8 3 6 2 5 - 4 t h - 8 t h + t h + 8 t h - 4 h t + 4 h t - 8 t h 2 10 3 17 7 2 14 2 12 3 13 + 6 t h - 4 t h - 4 t h + 4 t h + 8 t h ) (1 - 424 t h 3 11 3 15 4 20 2 8 4 24 2 4 - 320 t h - 380 t h + 662 t h + 112 t h + 144 t h + 24 t h 2 2 3 6 2 7 31 5 2 10 + 4 t h - 16 h t + 62 h t - 4 t h - 20 t h + 144 t h 4 12 4 26 7 35 7 33 6 32 + 33 t h - 4 h t + 64 t h - 20 t h - 16 t h + 96 t h 3 9 3 7 8 40 6 34 6 26 6 24 - 128 t h - 20 t h + t h + 24 t h + 62 t h + 24 t h 3 17 6 22 6 30 3 19 3 21 7 - 280 t h + 4 t h + 144 t h - 144 t h - 32 t h - 8 t h 6 28 5 23 5 25 2 14 5 17 2 12 + 112 t h - 424 t h - 380 t h + 24 t h - 20 t h + 96 t h 4 16 5 19 5 31 5 29 5 27 + 516 t h - 128 t h - 32 t h - 144 t h - 280 t h 5 21 7 37 4 22 4 14 4 28 4 18 - 320 t h - 8 t h + 352 t h + 200 t h + 16 t h + 752 t h )) and in Maple input format: (-h^2*(1+3*h^2+h^4)*(h^4+9*h^2+9)+h^3*(1369*h^8+290*h^2+901*h^4+160*h^12+12*h^ 14+38+1482*h^6+683*h^10)*t-h^4*(1+h^2)*(60*h^18+812*h^16+3838*h^14+9348*h^12+ 13569*h^10+12448*h^8+7300*h^6+2669*h^4+560*h^2+52)*t^2+h^5*(214903*h^12+2520*h^ 24+160*h^26+55553*h^8+16722*h^6+24+416*h^2+129240*h^10+16028*h^22+222031*h^16+ 257445*h^14+57982*h^20+3376*h^4+136379*h^18)*t^3-h^8*(1358216*h^18+1582719*h^16 +240*h^30+469510*h^10+138832*h^24+4160*h^28+7476*h^4+44413*h^6+934994*h^12+732* h^2+1397020*h^14+873735*h^20+172514*h^8+32+31224*h^26+413108*h^22)*t^4+h^13*(1+ h^2)*(192*h^30+3712*h^28+31824*h^26+161632*h^24+546780*h^22+1314526*h^20+ 2340365*h^18+3167760*h^16+3302566*h^14+2650771*h^12+1615142*h^10+727295*h^8+ 232583*h^6+49794*h^4+6430*h^2+384)*t^5-h^18*(1+h^2)*(64*h^32+1856*h^30+21280*h^ 28+136992*h^26+575924*h^24+1718192*h^22+3828954*h^20+6582857*h^18+8886233*h^16+ 9460802*h^14+7889083*h^12+5068414*h^10+2445678*h^8+853612*h^6+202930*h^4+29369* h^2+1952)*t^6+h^23*(5496+1018536*h^22+2612628*h^20+292784*h^24+3876692*h^8+ 72581*h^2+5153346*h^18+6976*h^28+434542*h^4+9353738*h^12+57888*h^26+6932564*h^ 10+1573249*h^6+9756369*h^14+7982091*h^16+384*h^30)*(1+h^2)^2*t^7-h^28*(1+h^2)*( 896*h^32+14848*h^30+115488*h^28+567792*h^26+1996288*h^24+5343380*h^22+11243718* h^20+18889635*h^18+25486087*h^16+27586166*h^14+23769980*h^12+16054287*h^10+ 8284514*h^8+3137134*h^6+816518*h^4+129667*h^2+9414)*t^8+h^33*(1+h^2)*(1024*h^32 +16320*h^30+123936*h^28+601008*h^26+2096240*h^24+5584692*h^22+11721814*h^20+ 19670011*h^18+26529668*h^16+28725766*h^14+24784507*h^12+16782126*h^10+8693377*h ^8+3308791*h^6+866686*h^4+138677*h^2+10154)*t^9-h^38*(1+h^2)*(576*h^32+9856*h^ 30+80928*h^28+418768*h^26+1524820*h^24+4156940*h^22+8814306*h^20+14847323*h^18+ 20029533*h^16+21611586*h^14+18497082*h^12+12370246*h^10+6309404*h^8+2361200*h^6 +608090*h^4+95724*h^2+6900)*t^10+h^43*(1+h^2)*(128*h^32+3136*h^30+31776*h^28+ 188432*h^26+752120*h^24+2174140*h^22+4758518*h^20+8110263*h^18+10926750*h^16+ 11671801*h^14+9812498*h^12+6390401*h^10+3146061*h^8+1128145*h^6+277171*h^4+ 41568*h^2+2856)*t^11-h^48*(3327213*h^10+11296*h^2+2590180*h^22+7446592*h^18+ 59472*h^28+4983921*h^20+1022056*h^24+1406554*h^8+8753727*h^16+295896*h^26+7472* h^30+429953*h^6+656+89384*h^4+448*h^32+5917899*h^12+8125516*h^14)*t^12+h^53*( 2132859*h^14+9056*h^28+1456*h^2+234308*h^24+655838*h^22+79990*h^6+1464932*h^12+ 14224*h^4+64+1339069*h^20+2048303*h^18+672*h^30+58160*h^26+763966*h^10+2385202* h^16+294390*h^8)*t^13-h^60*(1+h^2)*(560*h^24+5820*h^22+27896*h^20+81510*h^18+ 160623*h^16+222607*h^14+220448*h^12+156619*h^10+79599*h^8+28494*h^6+6860*h^4+ 992*h^2+64)*t^14+h^67*(41997*h^10+16+264*h^2+53361*h^12+28581*h^16+2668*h^20+ 8352*h^6+280*h^22+11358*h^18+47163*h^14+22835*h^8+1952*h^4)*t^15-h^76*(1068*h^4 +637*h^14+3633*h^10+84*h^16+24+2043*h^12+2621*h^6+244*h^2+3917*h^8)*t^16+2*h^85 *(2+h^2)*(7*h^2+3)*(1+h^2)^3*t^17-h^94*(2+h^2)*(1+h^2)*t^18)/(t*h^5-1)/(t^2*h^ 10+1-2*t*h^5-2*t*h^7)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-2*t^3*h^11+6*t^2*h^10+12* t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1)/(1-4*t^3*h^13-8*t^3*h^15+t^4*h^20+8* t^2*h^8-4*h^3*t+4*h^6*t^2-8*t*h^5+6*t^2*h^10-4*t^3*h^17-4*t*h^7+4*t^2*h^14+8*t^ 2*h^12)/(1-424*t^3*h^13-320*t^3*h^11-380*t^3*h^15+662*t^4*h^20+112*t^2*h^8+144* t^4*h^24+24*t^2*h^4+4*t^2*h^2-16*h^3*t+62*h^6*t^2-4*t^7*h^31-20*t*h^5+144*t^2*h ^10+33*t^4*h^12-4*h*t+64*t^4*h^26-20*t^7*h^35-16*t^7*h^33+96*t^6*h^32-128*t^3*h ^9-20*t^3*h^7+t^8*h^40+24*t^6*h^34+62*t^6*h^26+24*t^6*h^24-280*t^3*h^17+4*t^6*h ^22+144*t^6*h^30-144*t^3*h^19-32*t^3*h^21-8*t*h^7+112*t^6*h^28-424*t^5*h^23-380 *t^5*h^25+24*t^2*h^14-20*t^5*h^17+96*t^2*h^12+516*t^4*h^16-128*t^5*h^19-32*t^5* h^31-144*t^5*h^29-280*t^5*h^27-320*t^5*h^21-8*t^7*h^37+352*t^4*h^22+200*t^4*h^ 14+16*t^4*h^28+752*t^4*h^18) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /1764901 1624521 1013333 2 3219 3\ n |------- - ------- b + ------- b - ---- b | (1/b) \ 22440 3740 11220 1496 / that in floating-point is: n 66.8056926239638 36.4645094590774 and in Maple input format: HFloat(66.8056926239637647)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 15583 502897 b 107777 b 1289 b /3541 2184 468 2 56 3\ ----- + -------- - --------- + ------- + |---- + ---- b - --- b + --- b | n 5610 5610 5610 2805 \935 187 187 935 / and in Maple-input format: 15583/5610+502897/5610*b-107777/5610*b^2+1289/2805*b^3+(3541/935+2184/187*b-468 /187*b^2+56/935*b^3)*n and in floating point: 5.22164119915843 + 4.10557280859901 n The asymptotic expression for the variance, as a function of n is: 2 3 36273007 b 222736 7765187 b 185698 b - ---------- + ------ + ---------- - --------- 42075 8415 42075 42075 /105574 26128 15728 2 622 3\ + |------ - ----- b + ----- b - ----- b | n \42075 2805 8415 14025 / and in Maple-input format: -36273007/42075*b+222736/8415+7765187/42075*b^2-185698/42075*b^3+(105574/42075-\ 26128/2805*b+15728/8415*b^2-622/14025*b^3)*n and in floating point: 2.96536890504092 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 /755497810818251 [1, 3 + ----------------------------------------------- + |--------------- n \ 28160464640 395868535803137 593747659641319 2 28400798598787 3\ / 2 - --------------- b + --------------- b - -------------- b | / n , 15 402292352 2816046464 5632092928 / / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b 25143272 22000363 22000363 176002904 / + ------------------------------------------------------- + | n \ 109528284629096541 286954640515303311 430392155354112057 2 ------------------ - ------------------ b + ------------------ b 275972553472 19712325248 137986276736 20586997557395961 3\ / 2 - ----------------- b | / n , 105 275972553472 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 107998196881527861 1980614777716651269 424378414252057029 2 ------------------ - ------------------- b + ------------------ b 19712325248 9856162624 9856162624 20299343067550257 3\ / 2 - ----------------- b | / n ] 19712325248 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(755497810818251/28160464640-395868535803137/402292352*b+593747659641319/ 2816046464*b^2-28400798598787/5632092928*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 109528284629096541/275972553472-286954640515303311/19712325248*b+ 430392155354112057/137986276736*b^2-20586997557395961/275972553472*b^3)/n^2, 105+(-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752* b^3)/n+(107998196881527861/19712325248-1980614777716651269/9856162624*b+ 424378414252057029/9856162624*b^2-20299343067550257/19712325248*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.761238608831915 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 12.0047124948301 15 - ---------------- + ----------------, n 2 n 106.402545323854 184.604855321300 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 47788542835569 [------------------------------------------------------ + |- -------------- n \ 86241422960 1001579894699661 1502215360621407 2 17963882819019 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 49280813120 344965691840 172482845920 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 119464631934525 5007585548893905 7510619902940235 2 - --------------- + ---------------- b - ---------------- b 2156035574 2464040656 17248284592 89813974807395 3\ / 2 + -------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 210710464120926135 15456404634254124435 473109292840243905 2 - ------------------ + -------------------- b - ------------------ b 34496569184 68993138368 9856162624 9900745700744655 3\ / 2 + ---------------- b | / n ] 8624142296 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-47788542835569/86241422960+1001579894699661/49280813120*b-\ 1502215360621407/344965691840*b^2+17963882819019/172482845920*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-119464631934525/2156035574+5007585548893905/2464040656*b-\ 7510619902940235/17248284592*b^2+89813974807395/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-210710464120926135/34496569184+15456404634254124435/ 68993138368*b-473109292840243905/9856162624*b^2+9900745700744655/8624142296*b^3 )/n^2] and in floating-point it is: 0.00714279180461598 0.0360480831957141 0.714279180461598 3.95954221793470 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 502.750599341107 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 23, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 1, c[2] = 3, d[1] = 3, d[2] = 3 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 2 8 2 26 24 22 ) A(n, h) t = (-1 - 18 h + h (1 + h ) (32 h + 496 h + 3392 h / ----- n = 0 20 18 16 14 12 10 + 14690 h + 43886 h + 92079 h + 139001 h + 156664 h + 135196 h 8 6 4 2 6 15 2 + 88080 h + 40840 h + 12224 h + 2016 h + 128) t - 2 h (2 + h ) ( 28 26 24 22 20 18 128 h + 1552 h + 10912 h + 46652 h + 131696 h + 274204 h 16 14 12 10 8 + 454788 h + 606719 h + 629274 h + 489272 h + 277664 h 6 4 2 9 6 8 12 10 + 111888 h + 30496 h + 5056 h + 384) t - 108 h - 67 h - h - 15 h 4 3 2 2 2 4 2 - 71 h - 4 h (2 + 3 h ) (2 + h ) (1 + h ) (3 h + 8 h + 2) t 66 2 10 8 6 4 2 22 33 + h (1 + h ) (10 h + 62 h + 143 h + 153 h + 72 h + 12) t + 4 h 6 24 8 12 28 (129248 h + 384 + 35218 h + 341312 h + 899254 h + 928 h 16 4 14 26 2 18 + 781739 h + 33280 h + 949492 h + 7228 h + 5248 h + 516616 h 10 20 30 22 15 24 12 + 644952 h + 276242 h + 64 h + 115510 h ) t - 2 h (865791 h 16 20 6 8 10 + 352 + 884274 h + 368815 h + 122800 h + 318498 h + 601222 h 18 32 14 30 26 24 + 638350 h + 40 h + 979902 h + 608 h + 19144 h + 64225 h 2 22 28 4 12 21 10 + 5024 h + 171284 h + 4252 h + 32088 h ) t + 4 h (1176936 h 8 6 2 26 24 20 + 682024 h + 288992 h + 14592 h + 15104 h + 60662 h + 442752 h 30 22 28 14 12 + 1152 + 192 h + 183844 h + 2472 h + 1601684 h + 1551082 h 4 18 16 11 18 26 4 + 83552 h + 853398 h + 1311047 h ) t + 2 h (17936 h + 29696 h 12 10 20 30 8 32 + 829836 h + 572724 h + 350205 h + 512 h + 300544 h + 32 h 22 14 24 28 16 + 162972 h + 939744 h + 61072 h + 3824 h + 320 + 844593 h 18 6 2 10 11 16 2 + 606908 h + 114720 h + 4608 h ) t + 2 h (367546 h + 9216 h 24 4 18 6 10 12 + 3336 h + 48832 h + 189864 h + 152800 h + 491888 h + 577260 h 14 20 22 26 8 7 + 526951 h + 70860 h + 18780 h + 768 + 304 h + 321200 h ) t - 4 27 8 18 12 22 28 h (686880 h + 861336 h + 1564102 h + 185194 h + 2488 h 2 10 24 16 6 4 + 14592 h + 1186392 h + 61190 h + 1323503 h + 290464 h + 83744 h 26 20 14 30 13 7 + 15252 h + 446318 h + 1152 + 1615956 h + 192 h ) t - 2 h ( 12 2 18 4 6 14 22 97767 h + 2112 h + 12212 h + 10768 h + 33408 h + 68998 h + 280 h 10 16 20 8 5 30 2 + 98172 h + 34744 h + 192 + 2708 h + 69004 h ) t + 2 h (1 + h ) ( 30 28 26 24 22 20 18 16 h + 224 h + 1792 h + 9176 h + 31674 h + 78954 h + 152385 h 16 14 12 10 8 6 + 237703 h + 298520 h + 291572 h + 213840 h + 114756 h + 43872 h 4 2 14 2 12 8 2 + 11376 h + 1792 h + 128) t + h (1991 h + 8 + 3539 h + 144 h 6 14 4 16 10 18 2 5 + 2264 h + 704 h + 820 h + 128 h + 3364 h + 8 h ) t + 2 h ( 8 16 2 14 12 10 6 8227 h + 112 + 116 h + 960 h + 966 h + 3362 h + 6604 h + 6688 h 4 3 4 6 4 2 12 24 + 3400 h ) t - h (10244 h + 2476 h + 320 h + 49814 h + 24 h 10 8 16 14 20 22 + 43448 h + 25993 h + 16 + 22973 h + 40118 h + 2513 h + 392 h 18 4 56 2 12 4 6 + 9288 h ) t - h (464 h + 3925 h + 48 + 1913 h + 4444 h 14 16 8 10 18 20 49 20 + 1600 h + 381 h + 6470 h + 6190 h + 40 h ) t + 2 h (3040 h 22 10 4 14 16 + 320 h + 110104 h + 192 + 11872 h + 77704 h + 39166 h 12 18 2 6 8 19 38 + 109727 h + 13714 h + 2240 h + 37456 h + 77540 h ) t - h (256 6 12 10 24 20 14 + 73084 h + 374246 h + 290625 h + 6208 h + 79357 h + 376085 h 26 16 18 28 2 22 + 984 h + 294636 h + 176691 h + 80 h + 3472 h + 26197 h 4 8 16 41 12 2 10 + 20768 h + 171588 h ) t - 4 h (307596 h + 4672 h + 262114 h 20 18 16 4 22 + 37715 h + 384 + 100631 h + 194396 h + 25120 h + 9986 h 24 6 8 14 26 17 46 + 1760 h + 79784 h + 169896 h + 279521 h + 160 h ) t + h ( 6 16 22 24 10 4 2 17224 h + 35583 h + 888 h + 64 + 80 h + 63784 h + 4904 h + 832 h 12 18 14 20 8 18 12 + 72507 h + 15400 h + 59524 h + 4656 h + 39860 h ) t - h ( 10 6 20 30 8 22 565724 h + 90112 h + 399894 h + 304 h + 192 + 267812 h + 188803 h 24 12 14 28 32 26 + 67706 h + 877182 h + 1026232 h + 2904 h + 16 h + 17216 h 16 4 2 18 8 + 932387 h + 20528 h + 2880 h + 677348 h ) t 69 2 2 4 6 2 23 + 2 h (3 + 2 h ) (2 h + 1) (12 + 33 h + 10 h + 36 h ) t 79 2 2 25 59 2 2 - 2 h (3 + 2 h ) (2 h + 1) t - 2 h (3 + 2 h ) (2 h + 1) 12 10 8 6 4 2 21 (40 h + 230 h + 569 h + 788 h + 632 h + 272 h + 48) t 76 2 2 4 24 / 3 3 - h (1 + h ) (1 + 3 h + h ) t ) / ((h t - 1) (h t + 1) / 6 2 3 6 2 3 (h t + 1 + 2 h t + 2 h t) (h t + 1 - 2 h t - 2 h t) 4 12 6 2 2 8 (t h + 1 - 2 h t - 2 t h ) 2 8 2 4 3 6 2 4 12 3 9 (1 - 2 t h - 2 t h - 2 h t - 2 h t + t h - 2 t h ) 2 8 2 4 3 6 2 4 12 3 9 8 24 (1 - 2 t h - 2 t h + 2 h t - 2 h t + t h + 2 t h ) (t h 6 20 6 18 6 16 4 16 6 14 4 14 - 4 t h - 20 t h - 24 t h + 4 t h - 8 t h + 8 t h 4 12 4 10 4 8 2 8 4 6 6 2 + 6 t h + 32 t h + 80 t h - 4 t h + 64 t h - 20 h t 4 4 2 4 2 2 + 16 t h - 24 t h - 8 t h + 1)) and in Maple input format: (-1-18*h^2+h^8*(1+h^2)*(32*h^26+496*h^24+3392*h^22+14690*h^20+43886*h^18+92079* h^16+139001*h^14+156664*h^12+135196*h^10+88080*h^8+40840*h^6+12224*h^4+2016*h^2 +128)*t^6-2*h^15*(2+h^2)*(128*h^28+1552*h^26+10912*h^24+46652*h^22+131696*h^20+ 274204*h^18+454788*h^16+606719*h^14+629274*h^12+489272*h^10+277664*h^8+111888*h ^6+30496*h^4+5056*h^2+384)*t^9-108*h^6-67*h^8-h^12-15*h^10-71*h^4-4*h^3*(2+3*h^ 2)*(2+h^2)*(1+h^2)*(3*h^4+8*h^2+2)*t+h^66*(1+h^2)*(10*h^10+62*h^8+143*h^6+153*h ^4+72*h^2+12)*t^22+4*h^33*(129248*h^6+384+35218*h^24+341312*h^8+899254*h^12+928 *h^28+781739*h^16+33280*h^4+949492*h^14+7228*h^26+5248*h^2+516616*h^18+644952*h ^10+276242*h^20+64*h^30+115510*h^22)*t^15-2*h^24*(865791*h^12+352+884274*h^16+ 368815*h^20+122800*h^6+318498*h^8+601222*h^10+638350*h^18+40*h^32+979902*h^14+ 608*h^30+19144*h^26+64225*h^24+5024*h^2+171284*h^22+4252*h^28+32088*h^4)*t^12+4 *h^21*(1176936*h^10+682024*h^8+288992*h^6+14592*h^2+15104*h^26+60662*h^24+ 442752*h^20+1152+192*h^30+183844*h^22+2472*h^28+1601684*h^14+1551082*h^12+83552 *h^4+853398*h^18+1311047*h^16)*t^11+2*h^18*(17936*h^26+29696*h^4+829836*h^12+ 572724*h^10+350205*h^20+512*h^30+300544*h^8+32*h^32+162972*h^22+939744*h^14+ 61072*h^24+3824*h^28+320+844593*h^16+606908*h^18+114720*h^6+4608*h^2)*t^10+2*h^ 11*(367546*h^16+9216*h^2+3336*h^24+48832*h^4+189864*h^18+152800*h^6+491888*h^10 +577260*h^12+526951*h^14+70860*h^20+18780*h^22+768+304*h^26+321200*h^8)*t^7-4*h ^27*(686880*h^8+861336*h^18+1564102*h^12+185194*h^22+2488*h^28+14592*h^2+ 1186392*h^10+61190*h^24+1323503*h^16+290464*h^6+83744*h^4+15252*h^26+446318*h^ 20+1152+1615956*h^14+192*h^30)*t^13-2*h^7*(97767*h^12+2112*h^2+12212*h^18+10768 *h^4+33408*h^6+68998*h^14+280*h^22+98172*h^10+34744*h^16+192+2708*h^20+69004*h^ 8)*t^5+2*h^30*(1+h^2)*(16*h^30+224*h^28+1792*h^26+9176*h^24+31674*h^22+78954*h^ 20+152385*h^18+237703*h^16+298520*h^14+291572*h^12+213840*h^10+114756*h^8+43872 *h^6+11376*h^4+1792*h^2+128)*t^14+h^2*(1991*h^12+8+3539*h^8+144*h^2+2264*h^6+ 704*h^14+820*h^4+128*h^16+3364*h^10+8*h^18)*t^2+2*h^5*(8227*h^8+112+116*h^16+ 960*h^2+966*h^14+3362*h^12+6604*h^10+6688*h^6+3400*h^4)*t^3-h^4*(10244*h^6+2476 *h^4+320*h^2+49814*h^12+24*h^24+43448*h^10+25993*h^8+16+22973*h^16+40118*h^14+ 2513*h^20+392*h^22+9288*h^18)*t^4-h^56*(464*h^2+3925*h^12+48+1913*h^4+4444*h^6+ 1600*h^14+381*h^16+6470*h^8+6190*h^10+40*h^18)*t^20+2*h^49*(3040*h^20+320*h^22+ 110104*h^10+192+11872*h^4+77704*h^14+39166*h^16+109727*h^12+13714*h^18+2240*h^2 +37456*h^6+77540*h^8)*t^19-h^38*(256+73084*h^6+374246*h^12+290625*h^10+6208*h^ 24+79357*h^20+376085*h^14+984*h^26+294636*h^16+176691*h^18+80*h^28+3472*h^2+ 26197*h^22+20768*h^4+171588*h^8)*t^16-4*h^41*(307596*h^12+4672*h^2+262114*h^10+ 37715*h^20+384+100631*h^18+194396*h^16+25120*h^4+9986*h^22+1760*h^24+79784*h^6+ 169896*h^8+279521*h^14+160*h^26)*t^17+h^46*(17224*h^6+35583*h^16+888*h^22+64+80 *h^24+63784*h^10+4904*h^4+832*h^2+72507*h^12+15400*h^18+59524*h^14+4656*h^20+ 39860*h^8)*t^18-h^12*(565724*h^10+90112*h^6+399894*h^20+304*h^30+192+267812*h^8 +188803*h^22+67706*h^24+877182*h^12+1026232*h^14+2904*h^28+16*h^32+17216*h^26+ 932387*h^16+20528*h^4+2880*h^2+677348*h^18)*t^8+2*h^69*(3+2*h^2)*(2*h^2+1)*(12+ 33*h^4+10*h^6+36*h^2)*t^23-2*h^79*(3+2*h^2)*(2*h^2+1)*t^25-2*h^59*(3+2*h^2)*(2* h^2+1)*(40*h^12+230*h^10+569*h^8+788*h^6+632*h^4+272*h^2+48)*t^21-h^76*(1+h^2)* (1+3*h^2+h^4)*t^24)/(h^3*t-1)/(h^3*t+1)/(h^6*t^2+1+2*h*t+2*h^3*t)/(h^6*t^2+1-2* h*t-2*h^3*t)/(t^4*h^12+1-2*h^6*t^2-2*t^2*h^8)/(1-2*t^2*h^8-2*t^2*h^4-2*h^3*t-2* h^6*t^2+t^4*h^12-2*t^3*h^9)/(1-2*t^2*h^8-2*t^2*h^4+2*h^3*t-2*h^6*t^2+t^4*h^12+2 *t^3*h^9)/(t^8*h^24-4*t^6*h^20-20*t^6*h^18-24*t^6*h^16+4*t^4*h^16-8*t^6*h^14+8* t^4*h^14+6*t^4*h^12+32*t^4*h^10+80*t^4*h^8-4*t^2*h^8+64*t^4*h^6-20*h^6*t^2+16*t ^4*h^4-24*t^2*h^4-8*t^2*h^2+1) We now present a statistical analysis 1/2 1/2 3 6 5 2 Let, b, be the algebraic number, ------ - ------ 2 2 4 2 whose minimal polynomial is, b - 52 b + 1 and whose floating-point appx. , to, 10, digits is, 0.138700709 We have the following proven facts The total number of tilings of the region is asymptotic to: /2755 35113 53 2 1351 3\ n |---- + ----- b - -- b - ---- b | (1/b) \ 24 45 24 90 / that in floating-point is: n 222.9357508 7.209768481 and in Maple input format: 222.9357508*7.209768481^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 / 2 \ 102 b b 2 b |109 b | 43/5 - ----- + ---- + ---- + |--- + ----| n 5 15 5 \45 45 / and in Maple-input format: 43/5-102/5*b+1/15*b^2+2/5*b^3+(109/45+1/45*b^2)*n and in floating point: 5.772855385 + 2.422649731 n The asymptotic expression for the variance, as a function of n is: 2 3 / 2\ 214 2 b 6 b 306 b |52 2 b | - --- - ---- - ---- + ----- + |-- - ----| n 45 15 5 5 \45 45 / and in Maple-input format: -214/45-2/15*b^2-6/5*b^3+306/5*b+(52/45-2/45*b^2)*n and in floating point: 3.727160813 + 1.154700539 n The even alpha coefficients until the, 8, -th , are: 2 52 2 b 14579 2 3 - -- + ---- - ----- - 2/15 b - 93/2 b + 4743/2 b 45 45 45 [1, 3 + ----------- + --------------------------------------, n 2 n 2 2 b - 52/3 + ---- 2 3 3 - 14563/3 + 71145/2 b - 2 b - 1395/2 b 15 + ------------- + ----------------------------------------, n 2 n 2 28 b - 728/3 + ----- 2 3 3 - 203518/3 - 28 b - 9765 b + 498015 b 105 + --------------- + ---------------------------------------] n 2 n and in Maple-input format: [1, 3+(-52/45+2/45*b^2)/n+(-14579/45-2/15*b^2-93/2*b^3+4743/2*b)/n^2, 15+(-52/3 +2/3*b^2)/n+(-14563/3+71145/2*b-2*b^2-1395/2*b^3)/n^2, 105+(-728/3+28/3*b^2)/n+ (-203518/3-28*b^2-9765*b^3+498015*b)/n^2] and in floating-point it is: 1.154700539 4.8243122 17.32050807 77.69801704 [1, 3 - ----------- + ---------, 15 - ----------- + -----------, n 2 n 2 n n 242.4871131 1209.10557 105 - ----------- + ----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 24, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 1, d[1] = 1, d[2] = 1 Then infinity ----- 2 2 2 2 4 2 2 \ n -2 h - 1 + h (2 + h ) (1 + h ) t - h (1 + h ) t ) A(n, h) t = --------------------------------------------------- / 2 4 2 4 2 ----- (h t - 1) (-t h + t h - 2 h t + 1) n = 0 and in Maple input format: (-2*h^2-1+h^2*(2+h^2)*(1+h^2)*t-h^4*(1+h^2)*t^2)/(h^2*t-1)/(-t*h^4+t^2*h^4-2*h^ 2*t+1) We now present a statistical analysis 1/2 5 Let, b, be the algebraic number, 3/2 - ---- 2 2 whose minimal polynomial is, b - 3 b + 1 and whose floating-point appx. , to, 10, digits is, 0.381966012 We have the following proven facts The total number of tilings of the region is asymptotic to: / 4 b\ n |11/5 - ---| (1/b) \ 5 / that in floating-point is: n 1.894427190 2.618033984 and in Maple input format: 1.894427190*2.618033984^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 6 b / 4 b\ 8/5 - --- + |16/5 - ---| n 5 \ 5 / and in Maple-input format: 8/5-6/5*b+(16/5-4/5*b)*n and in floating point: 1.141640786 + 2.894427190 n The asymptotic expression for the variance, as a function of n is: 28 24 b /24 16 b\ -- - ---- + |-- - ----| n 25 25 \25 25 / and in Maple-input format: 28/25-24/25*b+(24/25-16/25*b)*n and in floating point: 0.7533126285 + 0.7155417523 n The even alpha coefficients until the, 8, -th , are: 23 3 b 39 b -- - --- 19/2 - ---- - 3/2 + b 20 2 13 b - 39/2 2 [1, 3 + --------- + --------, 15 + ----------- + -----------, n 2 n 2 n n -231 + 154 b - 119/4 - 231 b 105 + ------------ + ---------------] n 2 n and in Maple-input format: [1, 3+(-3/2+b)/n+(23/20-3/2*b)/n^2, 15+(13*b-39/2)/n+(19/2-39/2*b)/n^2, 105+(-\ 231+154*b)/n+(-119/4-231*b)/n^2] and in floating-point it is: 1.118033988 0.5770509820 14.53444184 2.051662766 [1, 3 - ----------- + ------------, 15 - ----------- + -----------, n 2 n 2 n n 172.1772342 117.9841488 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 3 b - 3/4 + --- 3/10 - b/5 10 30 - 20 b -110 + 30 b [---------- + -----------, --------- + -----------, n 2 n 2 n n 6615 b - 71295/4 + ------ 6615/2 - 2205 b 2 --------------- + ------------------] n 2 n and in Maple-input format: [(3/10-1/5*b)/n+(-3/4+3/10*b)/n^2, (30-20*b)/n+(-110+30*b)/n^2, (6615/2-2205*b) /n+(-71295/4+6615/2*b)/n^2] and in floating-point it is: 0.2236067976 0.6354101964 22.36067976 98.54101964 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 2465.264944 16560.39742 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 25, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 1, d[1] = 1, d[2] = 3 Then infinity ----- \ n 2 4 ) A(n, h) t = (1 + 6 h + 4 h / ----- n = 0 2 2 6 4 8 - h (27 h + 4 + 38 h + 51 h + 9 h ) t 4 4 2 6 8 10 12 2 + h (110 h + 4 + 36 h + 163 h + 127 h + 48 h + 7 h ) t 8 10 8 6 4 2 14 12 3 - h (80 h + 186 h + 250 h + 192 h + 76 h + 12 + 2 h + 19 h ) t 12 8 10 2 4 6 14 12 4 + h (154 h + 8 + 63 h + 52 h + 141 h + 201 h + h + 13 h ) t 18 2 8 6 4 2 5 - h (1 + h ) (3 h + 20 h + 43 h + 39 h + 12) t 24 2 4 2 6 30 2 7 / + h (1 + h ) (6 + 3 h + 10 h ) t - h (1 + h ) t ) / ( / 2 8 4 6 2 8 2 4 4 16 (t h + 1 - 2 t h - t h ) (t h + 1 - 2 h t - 2 t h ) (t h 3 14 3 12 2 12 3 10 2 10 2 8 6 2 - 2 t h - 6 t h + t h - 4 t h + 4 t h + 6 t h + 8 h t 6 2 4 4 2 - 2 t h + 4 t h - 6 t h - 4 h t + 1)) and in Maple input format: (1+6*h^2+4*h^4-h^2*(27*h^2+4+38*h^6+51*h^4+9*h^8)*t+h^4*(110*h^4+4+36*h^2+163*h ^6+127*h^8+48*h^10+7*h^12)*t^2-h^8*(80*h^10+186*h^8+250*h^6+192*h^4+76*h^2+12+2 *h^14+19*h^12)*t^3+h^12*(154*h^8+8+63*h^10+52*h^2+141*h^4+201*h^6+h^14+13*h^12) *t^4-h^18*(1+h^2)*(3*h^8+20*h^6+43*h^4+39*h^2+12)*t^5+h^24*(1+h^2)*(6+3*h^4+10* h^2)*t^6-h^30*(1+h^2)*t^7)/(t^2*h^8+1-2*t*h^4-t*h^6)/(t^2*h^8+1-2*h^2*t-2*t*h^4 )/(t^4*h^16-2*t^3*h^14-6*t^3*h^12+t^2*h^12-4*t^3*h^10+4*t^2*h^10+6*t^2*h^8+8*h^ 6*t^2-2*t*h^6+4*t^2*h^4-6*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /482 172 2 26 3\ 23/3 - ----- + ------ - ----- + |--- - 64/7 b + --- b - -- b | n 15 15 15 \105 21 35 / and in Maple-input format: 23/3-842/15*b+662/15*b^2-59/15*b^3+(482/105-64/7*b+172/21*b^2-26/35*b^3)*n and in floating point: 2.37964132686805 + 3.73972665214447 n The asymptotic expression for the variance, as a function of n is: 3 2 794308 b 25936 52546 b 593848 b - -------- + ----- - -------- + --------- 1575 525 1575 1575 / 416 656 2 92 3 2944\ + |- --- b + --- b - --- b + ----| n \ 105 315 525 1575/ and in Maple-input format: -794308/1575*b+25936/525-52546/1575*b^3+593848/1575*b^2+(-416/105*b+656/315*b^2 -92/525*b^3+2944/1575)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 26, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 1, d[1] = 2, d[2] = 2 Then infinity ----- \ n 2 4 ) A(n, h) t = (-1 - 6 h - 4 h / ----- n = 0 8 10 6 4 2 + (1 + 17 h + 3 h + 32 h + 27 h + 9 h ) t 2 2 12 10 8 6 4 2 2 - h (1 + h ) (h + 11 h + 32 h + 49 h + 34 h + 10 h + 1) t 6 2 4 10 8 6 4 2 3 + h (1 + 3 h + h ) (3 h + 9 h + 21 h + 28 h + 14 h + 2) t 10 10 4 12 14 8 2 6 4 - h (39 h + 31 h + 11 h + h + 1 + 64 h + 9 h + 57 h ) t 18 4 2 2 2 5 26 2 6 / 4 + h (2 h + 7 h + 2) (1 + h ) t - h (1 + h ) t ) / ((t h - 1) / 2 8 4 6 4 16 3 14 3 12 2 12 (t h + 1 - 2 t h - t h ) (t h - 2 t h - 8 t h + t h 3 10 2 10 3 8 2 8 6 2 6 2 4 - 9 t h + 4 t h - 2 t h + 6 t h + 16 h t - 2 t h + 20 t h 4 2 2 2 2 - 8 t h + 8 t h - 9 h t + t - 2 t + 1)) and in Maple input format: (-1-6*h^2-4*h^4+(1+17*h^8+3*h^10+32*h^6+27*h^4+9*h^2)*t-h^2*(1+h^2)*(h^12+11*h^ 10+32*h^8+49*h^6+34*h^4+10*h^2+1)*t^2+h^6*(1+3*h^2+h^4)*(3*h^10+9*h^8+21*h^6+28 *h^4+14*h^2+2)*t^3-h^10*(39*h^10+31*h^4+11*h^12+h^14+1+64*h^8+9*h^2+57*h^6)*t^4 +h^18*(2*h^4+7*h^2+2)*(1+h^2)^2*t^5-h^26*(1+h^2)*t^6)/(t*h^4-1)/(t^2*h^8+1-2*t* h^4-t*h^6)/(t^4*h^16-2*t^3*h^14-8*t^3*h^12+t^2*h^12-9*t^3*h^10+4*t^2*h^10-2*t^3 *h^8+6*t^2*h^8+16*h^6*t^2-2*t*h^6+20*t^2*h^4-8*t*h^4+8*t^2*h^2-9*h^2*t+t^2-2*t+ 1) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 9 - 4 5 2 whose minimal polynomial is, b - 18 b + 1 and whose floating-point appx. , to, 10, digits is, 0.055728092 We have the following proven facts The total number of tilings of the region is asymptotic to: /521 29 b\ n |--- - ----| (1/b) \50 50 / that in floating-point is: n 10.38767771 17.94427127 and in Maple input format: 10.38767771*17.94427127^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 57 3 b /31 b \ -- + --- + |-- + ----| n 20 20 \10 10 / and in Maple-input format: 57/20+3/20*b+(31/10+1/10*b)*n and in floating point: 2.858359214 + 3.105572809 n The asymptotic expression for the variance, as a function of n is: 49 7 b /54 6 b\ -- - --- + |-- - ---| n 25 25 \25 25 / and in Maple-input format: 49/25-7/25*b+(54/25-6/25*b)*n and in floating point: 1.944396134 + 2.146625258 n The even alpha coefficients until the, 8, -th , are: b 103 7 b 133 b 133 439 313 b - 3/8 + ---- --- - --- ----- - --- --- - ----- 24 360 144 216 24 108 432 [1, 3 + ------------ + ---------, 15 + ----------- + -----------, n 2 n 2 n n 917 917 b 4648 2177 b - --- + ----- ---- - ------ 12 108 81 216 105 + ------------- + -------------] n 2 n and in Maple-input format: [1, 3+(-3/8+1/24*b)/n+(103/360-7/144*b)/n^2, 15+(133/216*b-133/24)/n+(439/108-\ 313/432*b)/n^2, 105+(-917/12+917/108*b)/n+(4648/81-2177/216*b)/n^2] and in floating-point it is: 0.3726779962 0.2834021066 5.507352610 4.024437748 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 75.94349389 56.82104912 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 43 b 5 b 785 5 b 1/120 - ---- - ---- + ---- 5/6 - --- - --- + --- 1080 2160 2160 54 324 108 [------------ + -------------, --------- + -----------, n 2 n 2 n n 245 b 145775 245 b 735/8 - ----- - ------ + ----- 24 432 48 ------------- + ----------------] n 2 n and in Maple-input format: [(1/120-1/1080*b)/n+(-43/2160+1/2160*b)/n^2, (5/6-5/54*b)/n+(-785/324+5/108*b)/ n^2, (735/8-245/24*b)/n+(-145775/432+245/48*b)/n^2] and in floating-point it is: 0.008281733248 0.01988160737 0.8281733248 2.420259502 [-------------- - -------------, ------------ - -----------, n 2 n 2 n n 91.30610906 337.1576841 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 27, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 1, d[1] = 3, d[2] = 1 Then infinity ----- \ n 2 4 ) A(n, h) t = (1 + 6 h + 4 h / ----- n = 0 2 2 6 4 8 - h (27 h + 4 + 38 h + 51 h + 9 h ) t 4 4 2 6 8 10 12 2 + h (110 h + 4 + 36 h + 163 h + 127 h + 48 h + 7 h ) t 8 10 8 6 4 2 14 12 3 - h (80 h + 186 h + 250 h + 192 h + 76 h + 12 + 2 h + 19 h ) t 12 8 10 2 4 6 14 12 4 + h (154 h + 8 + 63 h + 52 h + 141 h + 201 h + h + 13 h ) t 18 2 8 6 4 2 5 - h (1 + h ) (3 h + 20 h + 43 h + 39 h + 12) t 24 2 4 2 6 30 2 7 / + h (1 + h ) (6 + 3 h + 10 h ) t - h (1 + h ) t ) / ( / 2 8 4 6 2 8 2 4 4 16 (t h + 1 - 2 t h - t h ) (t h + 1 - 2 h t - 2 t h ) (t h 3 14 3 12 2 12 3 10 2 10 2 8 6 2 - 2 t h - 6 t h + t h - 4 t h + 4 t h + 6 t h + 8 h t 6 2 4 4 2 - 2 t h + 4 t h - 6 t h - 4 h t + 1)) and in Maple input format: (1+6*h^2+4*h^4-h^2*(27*h^2+4+38*h^6+51*h^4+9*h^8)*t+h^4*(110*h^4+4+36*h^2+163*h ^6+127*h^8+48*h^10+7*h^12)*t^2-h^8*(80*h^10+186*h^8+250*h^6+192*h^4+76*h^2+12+2 *h^14+19*h^12)*t^3+h^12*(154*h^8+8+63*h^10+52*h^2+141*h^4+201*h^6+h^14+13*h^12) *t^4-h^18*(1+h^2)*(3*h^8+20*h^6+43*h^4+39*h^2+12)*t^5+h^24*(1+h^2)*(6+3*h^4+10* h^2)*t^6-h^30*(1+h^2)*t^7)/(t^2*h^8+1-2*t*h^4-t*h^6)/(t^2*h^8+1-2*h^2*t-2*t*h^4 )/(t^4*h^16-2*t^3*h^14-6*t^3*h^12+t^2*h^12-4*t^3*h^10+4*t^2*h^10+6*t^2*h^8+8*h^ 6*t^2-2*t*h^6+4*t^2*h^4-6*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /3503 2589 4217 2 59 3\ n |---- - ---- b + ---- b - -- b | (1/b) \420 140 420 70 / that in floating-point is: n 6.55204922926948 9.77063586195681 and in Maple input format: HFloat(6.55204922926948452)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 842 b 662 b 59 b /482 172 2 26 3\ 23/3 - ----- + ------ - ----- + |--- - 64/7 b + --- b - -- b | n 15 15 15 \105 21 35 / and in Maple-input format: 23/3-842/15*b+662/15*b^2-59/15*b^3+(482/105-64/7*b+172/21*b^2-26/35*b^3)*n and in floating point: 2.37964132686805 + 3.73972665214447 n The asymptotic expression for the variance, as a function of n is: 3 2 794308 b 25936 52546 b 593848 b /2944 416 656 2 92 3\ - -------- + ----- - -------- + --------- + |---- - --- b + --- b - --- b | n 1575 525 1575 1575 \1575 105 315 525 / and in Maple-input format: -794308/1575*b+25936/525-52546/1575*b^3+593848/1575*b^2+(2944/1575-416/105*b+ 656/315*b^2-92/525*b^3)*n and in floating point: 1.69955725315857 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 129556708746 4657983025235 3491295253550 2 618010253947 3 ------------ - ------------- b + ------------- b - ------------ b 10115 34391 34391 68782 + -------------------------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 / + ----------------------------------------------------- + | n \ 4817685604506 866058203885601 649147719248106 2 114908949400929 3 ------------- - --------------- b + --------------- b - --------------- b 34391 584647 584647 1169294 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 4913 4913 4913 4913 | / n , 105 + --------------------------------------------------------- / / n /103029205014081 1089467363398194 816469874441364 2 + |--------------- - ---------------- b + --------------- b \ 83521 83521 83521 72261915441993 3\ / 2 - -------------- b | / n ] 83521 / / and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 129556708746/10115-4657983025235/34391*b+3491295253550/34391*b^2-618010253947/ 68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+ 36770751/34391*b^3)/n+(4817685604506/34391-866058203885601/584647*b+ 649147719248106/584647*b^2-114908949400929/1169294*b^3)/n^2, 105+(-246835536/ 4913+2637282960/4913*b-2293010520/4913*b^2+207323694/4913*b^3)/n+( 103029205014081/83521-1089467363398194/83521*b+816469874441364/83521*b^2-\ 72261915441993/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.816719150740028 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 13.1869603497180 15 - ---------------- + ----------------, n 2 n 145.184702468171 214.995704501634 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 / [--------------------------------------------------- + | n \ 894688784028 160834806589452 120550506738912 2 10669601466834 3 - ------------ + --------------- b - --------------- b + -------------- b 171955 2923235 2923235 2923235 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b \ / 2 34391 34391 34391 34391 / | / n , --------------------------------------------------------- + | / / n \ 303951682272240 189069874876320 2409641993564640 2 - --------------- + --------------- b - ---------------- b 584647 34391 584647 213278017568880 3\ / 2 + --------------- b | / n , 584647 / / 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 81091512584849340 857583257497143660 643536800792670960 2 - ----------------- + ------------------ b - ------------------ b 1419857 1419857 1419857 56968118955379170 3\ / 2 + ----------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -894688784028/171955+160834806589452/2923235*b-120550506738912/2923235*b^2+ 10669601466834/2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-\ 7397956800/34391*b^2+668910960/34391*b^3)/n+(-303951682272240/584647+ 189069874876320/34391*b-2409641993564640/584647*b^2+213278017568880/584647*b^3) /n^2, (-12501125280/4913+133972120800/4913*b-116517819600/4913*b^2+10535347620/ 4913*b^3)/n+(-81091512584849340/1419857+857583257497143660/1419857*b-\ 643536800792670960/1419857*b^2+56968118955379170/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.0611662680398695 2.49983457428517 7.24778193084057 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 1051.99288475513 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 28, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 1, d[1] = 3, d[2] = 3 Then infinity ----- \ n 2 6 4 ) A(n, h) t = (-1 - 12 h - 8 h - 20 h / ----- n = 0 2 8 6 2 10 12 4 4 + h (505 h + 631 h + 108 h + 200 h + 8 + 31 h + 394 h ) t - h ( 16 6 12 14 4 2 18 493 h + 4336 h + 4967 h + 2069 h + 1532 h + 272 h + 51 h + 16 10 8 2 8 16 14 10 4 + 7466 h + 7185 h ) t + h (10184 h + 23435 h + 45672 h + 7664 h 20 22 12 2 6 8 + 128 + 548 h + 45 h + 38592 h + 1536 h + 21448 h + 38044 h 18 3 12 14 2 4 26 + 3027 h ) t - h (110774 h + 2304 h + 12720 h + 21 h + 192 6 10 24 12 22 8 + 41984 h + 135976 h + 294 h + 144010 h + 1977 h + 91060 h 16 18 20 4 18 12 8 + 63122 h + 26970 h + 8605 h ) t + h (247044 h + 183520 h 6 20 22 28 24 2 10 + 95936 h + 14242 h + 3399 h + 4 h + 579 h + 6912 h + 248584 h 26 16 4 18 14 5 24 + 66 h + 640 + 103394 h + 33408 h + 44161 h + 183770 h ) t - h 28 24 2 22 4 14 6 (3 h + 609 h + 7552 h + 3478 h + 36080 h + 186218 h + 102016 h 12 20 8 18 10 + 251318 h + 704 + 14336 h + 191780 h + 44545 h + 255632 h 26 16 6 30 10 4 8 + 66 h + 104673 h ) t + h (150780 h + 256 + 16384 h + 105672 h 28 2 26 20 22 18 12 + h + 3072 h + 26 h + 9210 h + 2123 h + 28374 h + 154332 h 24 6 16 14 7 38 16 + 314 h + 51376 h + 65746 h + 116160 h ) t - h (11521 h 18 12 2 24 14 4 + 3495 h + 256 + 44142 h + 2576 h + 5 h + 26497 h + 11344 h 10 22 6 8 20 8 46 2 + 53933 h + 85 h + 28860 h + 47480 h + 700 h ) t + h (1 + h ) ( 18 16 14 12 10 8 6 10 h + 120 h + 653 h + 2092 h + 4293 h + 5622 h + 4540 h 4 2 9 + 2184 h + 576 h + 64) t 56 8 6 4 2 2 3 10 - h (10 h + 65 h + 161 h + 160 h + 48) (1 + h ) t 66 2 2 2 2 11 76 2 12 / + h (5 h + 3) (1 + h ) (2 + h ) t - h (1 + h ) t ) / ( / 6 2 12 6 8 (t h - 1) (t h + 1 - 2 t h - t h ) 2 12 2 4 6 8 4 2 8 (t h + 1 - 4 h t - 8 t h - 2 t h ) (1 - 2 t h - 4 t h + 4 t h 4 24 2 10 2 14 2 16 3 20 3 18 3 16 + t h + 8 t h + 4 t h + t h - 2 t h - 6 t h - 4 t h 6 2 12 8 4 2 8 4 24 2 - 6 t h + 6 t h ) (1 - 2 t h - 20 t h + 80 t h + t h - 8 h t 2 4 6 2 2 10 2 14 2 16 3 20 + 16 t h + 64 h t + 32 t h + 4 t h + t h - 2 t h 3 18 3 16 6 2 12 3 14 - 12 t h - 20 t h - 12 t h + 6 t h - 8 t h )) and in Maple input format: (-1-12*h^2-8*h^6-20*h^4+h^2*(505*h^8+631*h^6+108*h^2+200*h^10+8+31*h^12+394*h^4 )*t-h^4*(493*h^16+4336*h^6+4967*h^12+2069*h^14+1532*h^4+272*h^2+51*h^18+16+7466 *h^10+7185*h^8)*t^2+h^8*(10184*h^16+23435*h^14+45672*h^10+7664*h^4+128+548*h^20 +45*h^22+38592*h^12+1536*h^2+21448*h^6+38044*h^8+3027*h^18)*t^3-h^12*(110774*h^ 14+2304*h^2+12720*h^4+21*h^26+192+41984*h^6+135976*h^10+294*h^24+144010*h^12+ 1977*h^22+91060*h^8+63122*h^16+26970*h^18+8605*h^20)*t^4+h^18*(247044*h^12+ 183520*h^8+95936*h^6+14242*h^20+3399*h^22+4*h^28+579*h^24+6912*h^2+248584*h^10+ 66*h^26+640+103394*h^16+33408*h^4+44161*h^18+183770*h^14)*t^5-h^24*(3*h^28+609* h^24+7552*h^2+3478*h^22+36080*h^4+186218*h^14+102016*h^6+251318*h^12+704+14336* h^20+191780*h^8+44545*h^18+255632*h^10+66*h^26+104673*h^16)*t^6+h^30*(150780*h^ 10+256+16384*h^4+105672*h^8+h^28+3072*h^2+26*h^26+9210*h^20+2123*h^22+28374*h^ 18+154332*h^12+314*h^24+51376*h^6+65746*h^16+116160*h^14)*t^7-h^38*(11521*h^16+ 3495*h^18+256+44142*h^12+2576*h^2+5*h^24+26497*h^14+11344*h^4+53933*h^10+85*h^ 22+28860*h^6+47480*h^8+700*h^20)*t^8+h^46*(1+h^2)*(10*h^18+120*h^16+653*h^14+ 2092*h^12+4293*h^10+5622*h^8+4540*h^6+2184*h^4+576*h^2+64)*t^9-h^56*(10*h^8+65* h^6+161*h^4+160*h^2+48)*(1+h^2)^3*t^10+h^66*(5*h^2+3)*(1+h^2)*(2+h^2)^2*t^11-h^ 76*(1+h^2)*t^12)/(t*h^6-1)/(t^2*h^12+1-2*t*h^6-t*h^8)/(t^2*h^12+1-4*h^2*t-8*t*h ^4-2*t*h^6)/(1-2*t*h^8-4*t*h^4+4*t^2*h^8+t^4*h^24+8*t^2*h^10+4*t^2*h^14+t^2*h^ 16-2*t^3*h^20-6*t^3*h^18-4*t^3*h^16-6*t*h^6+6*t^2*h^12)/(1-2*t*h^8-20*t*h^4+80* t^2*h^8+t^4*h^24-8*h^2*t+16*t^2*h^4+64*h^6*t^2+32*t^2*h^10+4*t^2*h^14+t^2*h^16-\ 2*t^3*h^20-12*t^3*h^18-20*t^3*h^16-12*t*h^6+6*t^2*h^12-8*t^3*h^14) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /598897 4419759 2757137 2 65689 3\ n |------ - ------- b + ------- b - ----- b | (1/b) \22440 29920 89760 89760 / that in floating-point is: n 22.6608598731619 36.4645094590774 and in Maple input format: HFloat(22.6608598731618898)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 273 41267 b 8917 b 427 b /13757 2192 1516 2 61 3\ --- - ------- + ------- - ------ + |----- - ---- b + ---- b - --- b | n 34 255 255 510 \2805 187 561 935 / and in Maple-input format: 273/34-41267/255*b+8917/255*b^2-427/510*b^3+(13757/2805-2192/187*b+1516/561*b^2 -61/935*b^3)*n and in floating point: 3.61764186686368 + 4.58502611425567 n The asymptotic expression for the variance, as a function of n is: 2 3 2073113 15412414 b 368597 b 71958134 b ------- + ----------- - --------- - ---------- 42075 42075 42075 42075 / 26128 15728 2 622 3 105574\ + |- ----- b + ----- b - ----- b + ------| n \ 2805 8415 14025 42075 / and in Maple-input format: 2073113/42075+15412414/42075*b^2-368597/42075*b^3-71958134/42075*b+(-26128/2805 *b+15728/8415*b^2-622/14025*b^3+105574/42075)*n and in floating point: 2.64580188025536 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 / [1, 3 + ----------------------------------------------- + | n \ 188873229085817 346388310500635 74219199907805 2 7100271025879 3\ --------------- - --------------- b + -------------- b - ------------- b | 3520058080 176002904 176002904 704011616 / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b / 2 25143272 22000363 22000363 176002904 / / n , 15 + ------------------------------------------------------- + | / n \ 54764097096363603 1004357290652248941 215199548380302861 2 ----------------- - ------------------- b + ------------------ b 68993138368 34496569184 34496569184 10293664984842231 3\ / 2 - ----------------- b | / n , 105 68993138368 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 53999462889597927 990332082992233677 212194469088606117 2 ----------------- - ------------------ b + ------------------ b 4928081312 2464040656 2464040656 1449988998989361 3\ / 2 - ---------------- b | / n ] 704011616 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(188873229085817/3520058080-346388310500635/176002904*b+74219199907805/ 176002904*b^2-7100271025879/704011616*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 54764097096363603/68993138368-1004357290652248941/34496569184*b+ 215199548380302861/34496569184*b^2-10293664984842231/68993138368*b^3)/n^2, 105+ (-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752*b^3) /n+(53999462889597927/4928081312-990332082992233677/2464040656*b+ 212194469088606117/2464040656*b^2-1449988998989361/704011616*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.657354281434682 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 10.6712005395274 15 - ---------------- + ----------------, n 2 n 106.402545323854 169.933484072189 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 382233603627279 [------------------------------------------------------ + |- --------------- n \ 344965691840 7010059103755707 1502010544350447 2 71845800648927 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 172482845920 172482845920 344965691840 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 955527986765295 17524112293071255 3754811065899105 2 - --------------- + ----------------- b - ---------------- b 8624142296 4312071148 4312071148 179604249480555 3\ / 2 + --------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 842699705777392635 15454832326232571135 473063578968942405 2 - ------------------ + -------------------- b - ------------------ b 68993138368 34496569184 4928081312 158396742930401235 3\ / 2 + ------------------ b | / n ] 68993138368 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-382233603627279/344965691840+7010059103755707/172482845920* b-1502010544350447/172482845920*b^2+71845800648927/344965691840*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-955527986765295/8624142296+17524112293071255/4312071148*b-\ 3754811065899105/4312071148*b^2+179604249480555/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-842699705777392635/68993138368+15454832326232571135/ 34496569184*b-473063578968942405/4928081312*b^2+158396742930401235/68993138368* b^3)/n^2] and in floating-point it is: 0.00714279180461598 0.0135674876180141 0.714279180461598 1.40858596551540 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 188.117654428738 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 29, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 1, d[2] = 1 Then infinity ----- \ n ) A(n, h) t = / ----- n = 0 2 4 2 2 4 2 4 2 2 1 + 3 h + h - h (1 + h ) (h + 4 h + 2) t + h (1 + h ) t -------------------------------------------------------------- 2 6 2 4 4 2 (h t - 1) (t h - t h + 4 t h + 2 h t - 1) and in Maple input format: (1+3*h^2+h^4-h^2*(1+h^2)*(h^4+4*h^2+2)*t+h^4*(1+h^2)*t^2)/(h^2*t-1)/(t*h^6-t^2* h^4+4*t*h^4+2*h^2*t-1) We now present a statistical analysis 1/2 3 5 Let, b, be the algebraic number, 7/2 - ------ 2 2 whose minimal polynomial is, b - 7 b + 1 and whose floating-point appx. , to, 10, digits is, 0.145898034 We have the following proven facts The total number of tilings of the region is asymptotic to: / 8 b\ n |11/3 - ---| (1/b) \ 15 / that in floating-point is: n 3.588854382 6.854101955 and in Maple input format: 3.588854382*6.854101955^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 4 b /58 8 b\ 12/5 - --- + |-- - ---| n 5 \15 15 / and in Maple-input format: 12/5-4/5*b+(58/15-8/15*b)*n and in floating point: 2.283281573 + 3.788854382 n The asymptotic expression for the variance, as a function of n is: 16 b /112 32 b\ 8/5 - ---- + |--- - ----| n 25 \75 75 / and in Maple-input format: 8/5-16/25*b+(112/75-32/75*b)*n and in floating point: 1.506625258 + 1.431083505 n The even alpha coefficients until the, 8, -th , are: 13 13 b 91 25 13 b -- - b/4 ---- - -- -- - ---- - 7/12 + b/6 40 6 12 16 4 [1, 3 + ------------ + --------, 15 + --------- + ---------, n 2 n 2 n n 77 b 609 77 b - 539/6 + ---- - --- - ---- 3 16 2 105 + -------------- + ------------] n 2 n and in Maple-input format: [1, 3+(-7/12+1/6*b)/n+(13/40-1/4*b)/n^2, 15+(13/6*b-91/12)/n+(25/16-13/4*b)/n^2 , 105+(-539/6+77/3*b)/n+(-609/16-77/2*b)/n^2] and in floating-point it is: 0.5590169943 0.2885254915 7.267220926 1.088331390 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 86.08861712 43.67957431 105 - ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 13 b 10 b 7/60 - ---- - -- + ---- 35/3 - ---- 30 40 20 3 - 165/4 + 5 b [----------- + -----------, ----------- + -------------, n 2 n 2 n n 735 b 2205 b 5145/4 - ----- - 47775/8 + ------ 2 4 -------------- + ------------------] n 2 n and in Maple-input format: [(7/60-1/30*b)/n+(-13/40+1/20*b)/n^2, (35/3-10/3*b)/n+(-165/4+5*b)/n^2, (5145/4 -735/2*b)/n+(-47775/8+2205/4*b)/n^2] and in floating-point it is: 0.1118033989 0.3177050983 11.18033989 40.52050983 [------------ - ------------, ----------- - -----------, n 2 n 2 n n 1232.632472 5891.448709 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 30, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 1, d[2] = 2 Then infinity ----- \ n 2 4 2 ) A(n, h) t = (-h (h + 6 h + 4) / ----- n = 0 3 6 2 10 4 8 + h (3 + 27 h + 17 h + h + 32 h + 9 h ) t 4 2 12 10 8 6 4 2 2 - h (1 + h ) (h + 10 h + 34 h + 49 h + 32 h + 11 h + 1) t 7 2 4 10 8 6 4 2 3 + h (1 + 3 h + h ) (2 h + 14 h + 28 h + 21 h + 9 h + 3) t 10 12 4 2 6 8 14 10 4 - h (9 h + 39 h + 1 + 11 h + 64 h + 57 h + h + 31 h ) t 15 4 2 2 2 5 20 2 6 / 3 + h (2 h + 7 h + 2) (1 + h ) t - h (1 + h ) t ) / ((h t - 1) / 6 2 3 2 14 3 13 4 12 2 12 (h t - 2 h t - h t + 1) (t h - 2 t h + t h + 8 t h 3 11 2 10 3 9 2 8 3 7 7 6 2 - 9 t h + 20 t h - 8 t h + 16 t h - 2 t h - 2 t h + 6 h t 5 2 4 3 2 2 - 9 t h + 4 t h - 8 h t + t h - 2 h t + 1)) and in Maple input format: (-h^2*(h^4+6*h^2+4)+h^3*(3+27*h^6+17*h^2+h^10+32*h^4+9*h^8)*t-h^4*(1+h^2)*(h^12 +10*h^10+34*h^8+49*h^6+32*h^4+11*h^2+1)*t^2+h^7*(1+3*h^2+h^4)*(2*h^10+14*h^8+28 *h^6+21*h^4+9*h^2+3)*t^3-h^10*(9*h^12+39*h^4+1+11*h^2+64*h^6+57*h^8+h^14+31*h^ 10)*t^4+h^15*(2*h^4+7*h^2+2)*(1+h^2)^2*t^5-h^20*(1+h^2)*t^6)/(h^3*t-1)/(h^6*t^2 -2*h^3*t-h*t+1)/(t^2*h^14-2*t^3*h^13+t^4*h^12+8*t^2*h^12-9*t^3*h^11+20*t^2*h^10 -8*t^3*h^9+16*t^2*h^8-2*t^3*h^7-2*t*h^7+6*h^6*t^2-9*t*h^5+4*t^2*h^4-8*h^3*t+t^2 *h^2-2*h*t+1) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 9 - 4 5 2 whose minimal polynomial is, b - 18 b + 1 and whose floating-point appx. , to, 10, digits is, 0.055728092 We have the following proven facts The total number of tilings of the region is asymptotic to: /521 29 b\ n |--- - ----| (1/b) \50 50 / that in floating-point is: n 10.38767771 17.94427127 and in Maple input format: 10.38767771*17.94427127^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 63 3 b /39 b \ -- - --- + |-- - ----| n 20 20 \10 10 / and in Maple-input format: 63/20-3/20*b+(39/10-1/10*b)*n and in floating point: 3.141640786 + 3.894427191 n The asymptotic expression for the variance, as a function of n is: 49 7 b /54 6 b\ -- - --- + |-- - ---| n 25 25 \25 25 / and in Maple-input format: 49/25-7/25*b+(54/25-6/25*b)*n and in floating point: 1.944396134 + 2.146625258 n The even alpha coefficients until the, 8, -th , are: b 103 7 b 133 b 133 439 313 b - 3/8 + ---- --- - --- ----- - --- --- - ----- 24 360 144 216 24 108 432 [1, 3 + ------------ + ---------, 15 + ----------- + -----------, n 2 n 2 n n 917 917 b 4648 2177 b - --- + ----- ---- - ------ 12 108 81 216 105 + ------------- + -------------] n 2 n and in Maple-input format: [1, 3+(-3/8+1/24*b)/n+(103/360-7/144*b)/n^2, 15+(133/216*b-133/24)/n+(439/108-\ 313/432*b)/n^2, 105+(-917/12+917/108*b)/n+(4648/81-2177/216*b)/n^2] and in floating-point it is: 0.3726779962 0.2834021066 5.507352610 4.024437748 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 75.94349389 56.82104912 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 43 b 5 b 785 5 b 1/120 - ---- - ---- + ---- 5/6 - --- - --- + --- 1080 2160 2160 54 324 108 [------------ + -------------, --------- + -----------, n 2 n 2 n n 245 b 145775 245 b 735/8 - ----- - ------ + ----- 24 432 48 ------------- + ----------------] n 2 n and in Maple-input format: [(1/120-1/1080*b)/n+(-43/2160+1/2160*b)/n^2, (5/6-5/54*b)/n+(-785/324+5/108*b)/ n^2, (735/8-245/24*b)/n+(-145775/432+245/48*b)/n^2] and in floating-point it is: 0.008281733248 0.01988160737 0.8281733248 2.420259502 [-------------- - -------------, ------------ - -----------, n 2 n 2 n n 91.30610906 337.1576841 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 31, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 1, d[2] = 3 Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = (-(h + 7 h + 1) (1 + h ) / ----- n = 0 2 2 12 10 8 6 4 2 4 + h (1 + h ) (2 h + 21 h + 73 h + 109 h + 90 h + 36 h + 4) t - h 4 2 6 16 14 20 18 8 (207 h + 4 + 44 h + 584 h + 96 h + 345 h + h + 15 h + 1027 h 10 12 2 8 8 10 14 20 + 1122 h + 774 h ) t + h (2021 h + 2504 h + 1381 h + 18 h 2 22 6 4 16 12 18 3 + 136 h + h + 1228 h + 526 h + 549 h + 2255 h + 16 + 134 h ) t 12 8 12 10 20 4 18 - h (2236 h + 2580 h + 20 + 2776 h + 32 h + 583 h + 212 h 22 2 16 6 14 4 16 12 + 2 h + 160 h + 770 h + 1348 h + 1724 h ) t + h (1307 h 14 22 4 2 6 8 10 20 + 735 h + h + 296 h + 68 h + 798 h + 1397 h + 1634 h + 14 h 18 16 5 22 + 84 h + 8 + 299 h ) t - h ( 14 4 16 10 12 6 2 8 32 h + 222 h + 3 h + 305 h + 135 h + 383 h + 77 h + 12 + 426 h ) 6 28 2 4 2 4 2 7 34 2 2 8 t + h (1 + h ) (3 h + 5 h + 3) (h + 4 h + 2) t - h (1 + h ) t ) / 4 2 8 4 6 8 / ((t h - 1) (t h + 1 - 2 t h - 4 t h - t h ) / 2 8 2 4 4 16 3 16 2 16 3 14 (t h + 1 - 2 h t - 2 t h ) (t h - 2 t h + t h - 10 t h 2 14 3 12 2 12 3 10 2 10 2 8 + 8 t h - 12 t h + 20 t h - 4 t h + 16 t h + 6 t h 8 6 2 6 2 4 4 2 - 2 t h + 8 h t - 10 t h + 4 t h - 12 t h - 4 h t + 1)) and in Maple input format: (-(h^4+7*h^2+1)*(1+h^2)^2+h^2*(1+h^2)*(2*h^12+21*h^10+73*h^8+109*h^6+90*h^4+36* h^2+4)*t-h^4*(207*h^4+4+44*h^2+584*h^6+96*h^16+345*h^14+h^20+15*h^18+1027*h^8+ 1122*h^10+774*h^12)*t^2+h^8*(2021*h^8+2504*h^10+1381*h^14+18*h^20+136*h^2+h^22+ 1228*h^6+526*h^4+549*h^16+2255*h^12+16+134*h^18)*t^3-h^12*(2236*h^8+2580*h^12+ 20+2776*h^10+32*h^20+583*h^4+212*h^18+2*h^22+160*h^2+770*h^16+1348*h^6+1724*h^ 14)*t^4+h^16*(1307*h^12+735*h^14+h^22+296*h^4+68*h^2+798*h^6+1397*h^8+1634*h^10 +14*h^20+84*h^18+8+299*h^16)*t^5-h^22*(32*h^14+222*h^4+3*h^16+305*h^10+135*h^12 +383*h^6+77*h^2+12+426*h^8)*t^6+h^28*(1+h^2)*(3*h^4+5*h^2+3)*(h^4+4*h^2+2)*t^7- h^34*(1+h^2)^2*t^8)/(t*h^4-1)/(t^2*h^8+1-2*t*h^4-4*t*h^6-t*h^8)/(t^2*h^8+1-2*h^ 2*t-2*t*h^4)/(t^4*h^16-2*t^3*h^16+t^2*h^16-10*t^3*h^14+8*t^2*h^14-12*t^3*h^12+ 20*t^2*h^12-4*t^3*h^10+16*t^2*h^10+6*t^2*h^8-2*t*h^8+8*h^6*t^2-10*t*h^6+4*t^2*h ^4-12*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 28 _Z + 63 _Z - 28 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 28 b + 63 b - 28 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0390932603206857 We have the following proven facts The total number of tilings of the region is asymptotic to: /63647 17041 7687 2 1237 3\ n |----- - ----- b + ---- b - ---- b | (1/b) \1980 220 220 990 / that in floating-point is: n 29.1701460305608 25.5798567783016 and in Maple input format: HFloat(29.1701460305607725)*HFloat(25.5798567783015649)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 1721 8996 b 9484 b 119 b /716 256 184 2 34 3\ ---- + ------ - ------- + ------ + |--- + --- b - --- b + --- b | n 495 495 495 165 \165 33 33 165 / and in Maple-input format: 1721/495+8996/495*b-9484/495*b^2+119/165*b^3+(716/165+256/33*b-184/33*b^2+34/ 165*b^3)*n and in floating point: 4.15800012753731 + 4.63415384322664 n The asymptotic expression for the variance, as a function of n is: 3 2 509272 b 1456 11698 b 917648 b /17416 1856 608 2 316 3\ -------- + ---- + -------- - --------- + |----- - ---- b + --- b - ---- b | n 7425 7425 2475 7425 \7425 495 495 7425 / and in Maple-input format: 509272/7425*b+1456/7425+11698/2475*b^3-917648/7425*b^2+(17416/7425-1856/495*b+ 608/495*b^2-316/7425*b^3)*n and in floating point: 2.68885841619758 + 2.20088386493014 n The even alpha coefficients until the, 8, -th , are: 1434034 338320 44840 2 6641 3 - ------- + ------- b + ------- b - ------- b 3110217 1036739 1036739 3110217 / 21969685269856 [1, 3 + ----------------------------------------------- + |- -------------- n \ 4774183095 120652827161038 211191088530932 2 5381346012815 3\ / 2 + --------------- b - --------------- b + ------------- b | / n , 15 954836619 954836619 636557746 / / 230378218 1170459120 676704360 2 24690643 3 - --------- + ---------- b - --------- b + -------- b 28934443 28934443 28934443 28934443 / + ------------------------------------------------------- + | n \ 6957636763169604 191048766088465162 334457385536666108 2 - ---------------- + ------------------ b - ------------------ b 97711614011 97711614011 97711614011 25566937383188123 3\ / 2 + ----------------- b | / n , 105 195423228022 / / 40139100484 338689918560 211315777680 2 7747433134 3 - ----------- + ------------ b - ------------ b + ---------- b 318278873 318278873 318278873 318278873 / + ---------------------------------------------------------------- + | n \ 100381232738332861 2756365931861602956 4826233409563180104 2 - ------------------ + ------------------- b - ------------------- b 97711614011 97711614011 97711614011 184466688465337117 3\ / 2 + ------------------ b | / n ] 97711614011 / / and in Maple-input format: [1, 3+(-1434034/3110217+338320/1036739*b+44840/1036739*b^2-6641/3110217*b^3)/n+ (-21969685269856/4774183095+120652827161038/954836619*b-211191088530932/ 954836619*b^2+5381346012815/636557746*b^3)/n^2, 15+(-230378218/28934443+ 1170459120/28934443*b-676704360/28934443*b^2+24690643/28934443*b^3)/n+(-\ 6957636763169604/97711614011+191048766088465162/97711614011*b-\ 334457385536666108/97711614011*b^2+25566937383188123/195423228022*b^3)/n^2, 105 +(-40139100484/318278873+338689918560/318278873*b-211315777680/318278873*b^2+ 7747433134/318278873*b^3)/n+(-100381232738332861/97711614011+ 2756365931861602956/97711614011*b-4826233409563180104/97711614011*b^2+ 184466688465337117/97711614011*b^3)/n^2] and in floating-point it is: 0.448248696679244 0.521390865623671 [1, 3 - ----------------- + -----------------, n 2 n 6.41636251823189 7.17009859652749 15 - ---------------- + ----------------, n 2 n 85.5259242078518 94.8577919234394 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 166459104 1131708672 765023616 2 140895504 3 - ---------- + ---------- b - --------- b + ---------- b 1591394365 318278873 318278873 1591394365 / 9690578062688 [---------------------------------------------------------- + |- ------------- n \ 44414370005 265995727357136 467337563593024 2 17863811162032 3\ / 2 + --------------- b - --------------- b + -------------- b | / n , 44414370005 44414370005 44414370005 / / 3329182080 113170867200 76502361600 2 2817910080 3 - ---------- + ------------ b - ----------- b + ---------- b 318278873 318278873 318278873 318278873 / -------------------------------------------------------------- + | n \ 193611642474400 5314424312751520 9344271811167680 2 - --------------- + ---------------- b - ---------------- b 8882874001 8882874001 8882874001 357187671073040 3\ / 2 + --------------- b | / n , 8882874001 / / 367042324320 12477088108800 8434385366400 2 310674586320 3 - ------------ + -------------- b - ------------- b + ------------ b 318278873 318278873 318278873 318278873 / ---------------------------------------------------------------------- + | n \ 6546032251382902320 179674484780934480960 316182229067425724640 2 - ------------------- + --------------------- b - --------------------- b 2727042318307 2727042318307 2727042318307 12086391703848061320 3\ / 2 + -------------------- b | / n ] 2727042318307 / / and in Maple-input format: [(-166459104/1591394365+1131708672/318278873*b-765023616/318278873*b^2+ 140895504/1591394365*b^3)/n+(-9690578062688/44414370005+265995727357136/ 44414370005*b-467337563593024/44414370005*b^2+17863811162032/44414370005*b^3)/n ^2, (-3329182080/318278873+113170867200/318278873*b-76502361600/318278873*b^2+ 2817910080/318278873*b^3)/n+(-193611642474400/8882874001+5314424312751520/ 8882874001*b-9344271811167680/8882874001*b^2+357187671073040/8882874001*b^3)/n^ 2, (-367042324320/318278873+12477088108800/318278873*b-8434385366400/318278873* b^2+310674586320/318278873*b^3)/n+(-6546032251382902320/2727042318307+ 179674484780934480960/2727042318307*b-316182229067425724640/2727042318307*b^2+ 12086391703848061320/2727042318307*b^3)/n^2] and in floating-point it is: 0.0307367931956769 0.114706617065132 3.07367931956768 12.7009262232392 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 338.873145154837 1637.18096304436 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 32, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 2, d[2] = 1 Then infinity ----- \ n 2 4 2 ) A(n, h) t = (-h (h + 6 h + 4) / ----- n = 0 3 6 2 10 4 8 + h (3 + 27 h + 17 h + h + 32 h + 9 h ) t 4 2 12 10 8 6 4 2 2 - h (1 + h ) (h + 10 h + 34 h + 49 h + 32 h + 11 h + 1) t 7 2 4 10 8 6 4 2 3 + h (1 + 3 h + h ) (2 h + 14 h + 28 h + 21 h + 9 h + 3) t 10 12 4 2 6 8 14 10 4 - h (9 h + 39 h + 1 + 11 h + 64 h + 57 h + h + 31 h ) t 15 4 2 2 2 5 20 2 6 / 3 + h (2 h + 7 h + 2) (1 + h ) t - h (1 + h ) t ) / ((h t - 1) / 6 2 3 2 14 3 13 4 12 2 12 (h t - 2 h t - h t + 1) (t h - 2 t h + t h + 8 t h 3 11 2 10 3 9 2 8 3 7 7 6 2 - 9 t h + 20 t h - 8 t h + 16 t h - 2 t h - 2 t h + 6 h t 5 2 4 3 2 2 - 9 t h + 4 t h - 8 h t + t h - 2 h t + 1)) and in Maple input format: (-h^2*(h^4+6*h^2+4)+h^3*(3+27*h^6+17*h^2+h^10+32*h^4+9*h^8)*t-h^4*(1+h^2)*(h^12 +10*h^10+34*h^8+49*h^6+32*h^4+11*h^2+1)*t^2+h^7*(1+3*h^2+h^4)*(2*h^10+14*h^8+28 *h^6+21*h^4+9*h^2+3)*t^3-h^10*(9*h^12+39*h^4+1+11*h^2+64*h^6+57*h^8+h^14+31*h^ 10)*t^4+h^15*(2*h^4+7*h^2+2)*(1+h^2)^2*t^5-h^20*(1+h^2)*t^6)/(h^3*t-1)/(h^6*t^2 -2*h^3*t-h*t+1)/(t^2*h^14-2*t^3*h^13+t^4*h^12+8*t^2*h^12-9*t^3*h^11+20*t^2*h^10 -8*t^3*h^9+16*t^2*h^8-2*t^3*h^7-2*t*h^7+6*h^6*t^2-9*t*h^5+4*t^2*h^4-8*h^3*t+t^2 *h^2-2*h*t+1) We now present a statistical analysis 1/2 Let, b, be the algebraic number, 9 - 4 5 2 whose minimal polynomial is, b - 18 b + 1 and whose floating-point appx. , to, 10, digits is, 0.055728092 We have the following proven facts The total number of tilings of the region is asymptotic to: /521 29 b\ n |--- - ----| (1/b) \50 50 / that in floating-point is: n 10.38767771 17.94427127 and in Maple input format: 10.38767771*17.94427127^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 63 3 b /39 b \ -- - --- + |-- - ----| n 20 20 \10 10 / and in Maple-input format: 63/20-3/20*b+(39/10-1/10*b)*n and in floating point: 3.141640786 + 3.894427191 n The asymptotic expression for the variance, as a function of n is: 49 7 b /54 6 b\ -- - --- + |-- - ---| n 25 25 \25 25 / and in Maple-input format: 49/25-7/25*b+(54/25-6/25*b)*n and in floating point: 1.944396134 + 2.146625258 n The even alpha coefficients until the, 8, -th , are: b 103 7 b 133 b 133 439 313 b - 3/8 + ---- --- - --- ----- - --- --- - ----- 24 360 144 216 24 108 432 [1, 3 + ------------ + ---------, 15 + ----------- + -----------, n 2 n 2 n n 917 917 b 4648 2177 b - --- + ----- ---- - ------ 12 108 81 216 105 + ------------- + -------------] n 2 n and in Maple-input format: [1, 3+(-3/8+1/24*b)/n+(103/360-7/144*b)/n^2, 15+(133/216*b-133/24)/n+(439/108-\ 313/432*b)/n^2, 105+(-917/12+917/108*b)/n+(4648/81-2177/216*b)/n^2] and in floating-point it is: 0.3726779962 0.2834021066 5.507352610 4.024437748 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 75.94349389 56.82104912 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: b 43 b 5 b 785 5 b 1/120 - ---- - ---- + ---- 5/6 - --- - --- + --- 1080 2160 2160 54 324 108 [------------ + -------------, --------- + -----------, n 2 n 2 n n 245 b 145775 245 b 735/8 - ----- - ------ + ----- 24 432 48 ------------- + ----------------] n 2 n and in Maple-input format: [(1/120-1/1080*b)/n+(-43/2160+1/2160*b)/n^2, (5/6-5/54*b)/n+(-785/324+5/108*b)/ n^2, (735/8-245/24*b)/n+(-145775/432+245/48*b)/n^2] and in floating-point it is: 0.008281733248 0.01988160737 0.8281733248 2.420259502 [-------------- - -------------, ------------ - -----------, n 2 n 2 n n 91.30610906 337.1576841 ----------- - -----------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 33, : Let A(n,h) be the weight-enumerator of of the set of domino tilings of the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 2, c[2] = 2, d[1] = 2, d[2] = 2 where the weight of a tiling is h raised to the power the number of horizontal tiles. Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = (-(h + 7 h + 1) (1 + h ) / ----- n = 0 8 12 4 2 10 6 + (1 - 22 h + h - 22 h + h + h - 48 h ) t 2 6 2 4 8 12 10 2 + h (12 h + 31 h + 7 + 32 h + 32 h + 7 h + 31 h ) t - 2 4 2 16 14 12 10 8 6 3 h (-39 h + 1 - 4 h + h - 4 h - 39 h - 78 h - 84 h - 78 h ) t 4 - h 16 4 8 6 2 12 10 14 4 (2 h + 34 h + 36 h + 45 h + 13 h + 34 h + 2 + 45 h + 13 h ) t - 6 6 10 2 16 4 8 14 12 5 h (77 h + 77 h + 13 h + h + 51 h + 68 h + 1 + 13 h + 51 h ) t 10 12 2 6 8 4 10 6 - h (3 h + 8 h + 3 - 70 h - 26 h - 26 h + 8 h ) t 14 8 6 4 2 7 18 2 4 8 + h (5 h + 5 + 31 h + 48 h + 31 h ) t + h (2 h + 3 h + 3) t 9 22 / 2 2 2 4 2 4 - 4 t h ) / ((h t + 1) (h t - 1) (t h + 1 + 2 h t + t h ) / 2 4 2 4 8 3 8 2 8 3 6 6 2 (t h + t + 1 + 2 h t) (t h - 2 t h + t h - 5 t h + 4 h t 3 4 2 4 4 2 2 2 2 - 2 t h + 6 t h - 2 t h + 4 t h - 5 h t + t - 2 t + 1)) and in Maple input format: (-(h^4+7*h^2+1)*(1+h^2)^2+(1-22*h^8+h^12-22*h^4+h^2+h^10-48*h^6)*t+h^2*(12*h^6+ 31*h^2+7+32*h^4+32*h^8+7*h^12+31*h^10)*t^2-h^2*(-39*h^4+1-4*h^2+h^16-4*h^14-39* h^12-78*h^10-84*h^8-78*h^6)*t^3-h^4*(2*h^16+34*h^4+36*h^8+45*h^6+13*h^2+34*h^12 +2+45*h^10+13*h^14)*t^4-h^6*(77*h^6+77*h^10+13*h^2+h^16+51*h^4+68*h^8+1+13*h^14 +51*h^12)*t^5-h^10*(3*h^12+8*h^2+3-70*h^6-26*h^8-26*h^4+8*h^10)*t^6+h^14*(5*h^8 +5+31*h^6+48*h^4+31*h^2)*t^7+h^18*(2*h^2+3*h^4+3)*t^8-4*t^9*h^22)/(h^2*t+1)/(h^ 2*t-1)/(t^2*h^4+1+2*h^2*t+t*h^4)/(t^2*h^4+t+1+2*h^2*t)/(t^4*h^8-2*t^3*h^8+t^2*h ^8-5*t^3*h^6+4*h^6*t^2-2*t^3*h^4+6*t^2*h^4-2*t*h^4+4*t^2*h^2-5*h^2*t+t^2-2*t+1) We now present a statistical analysis 1/2 3 5 Let, b, be the algebraic number, 7/2 - ------ 2 2 whose minimal polynomial is, b - 7 b + 1 and whose floating-point appx. , to, 10, digits is, 0.145898034 We have the following proven facts The total number of tilings of the region is asymptotic to: /768 112 b\ n |--- - -----| (1/b) \25 25 / that in floating-point is: n 30.06637681 6.854101955 and in Maple input format: 30.06637681*6.854101955^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 4 + 2 n and in Maple-input format: 4+2*n and in floating point: 4 + 2 n The asymptotic expression for the variance, as a function of n is: 188 64 b /112 32 b\ --- - ---- + |--- - ----| n 75 75 \75 75 / and in Maple-input format: 188/75-64/75*b+(112/75-32/75*b)*n and in floating point: 2.382167011 + 1.431083505 n The even alpha coefficients until the, 8, -th , are: 497 5 b 495 --- - b/3 --- - 35/4 --- - 5 b - 7/12 + b/6 480 2 32 [1, 3 + ------------ + ---------, 15 + ---------- + ---------, n 2 n 2 n n - 245/2 + 35 b 1813/8 - 70 b 105 + -------------- + -------------] n 2 n and in Maple-input format: [1, 3+(-7/12+1/6*b)/n+(497/480-1/3*b)/n^2, 15+(5/2*b-35/4)/n+(495/32-5*b)/n^2, 105+(-245/2+35*b)/n+(1813/8-70*b)/n^2] and in floating-point it is: 0.5590169943 0.9867839890 8.385254915 14.73925983 [1, 3 - ------------ + ------------, 15 - ----------- + -----------, n 2 n 2 n n 117.3935688 216.4121376 105 - ----------- + -----------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: [0, 0, 0] and in Maple-input format: [0, 0, 0] and in floating-point it is: [0, 0, 0] Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 34, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 2, d[2] = 3 Then infinity ----- \ n 2 2 4 4 2 3 12 ) A(n, h) t = (-h (1 + 3 h + h ) (h + 9 h + 9) + h (441 h / ----- n = 0 10 4 14 16 6 8 2 + 1208 h + 789 h + 79 h + 5 h + 1584 h + 1823 h + 207 h + 22) t 4 14 8 4 2 12 22 - h (16 + 32388 h + 21182 h + 2060 h + 284 h + 40909 h + 193 h 18 20 24 10 6 16 2 7 + 6479 h + 1501 h + 10 h + 35668 h + 8336 h + 17669 h ) t + h 26 16 18 10 30 14 (2217 h + 464091 h + 298063 h + 304953 h + 10 h + 541374 h 24 20 2 12 22 4 + 12909 h + 143036 h + 1668 h + 471896 h + 50714 h + 11390 h 28 6 8 3 10 12 6 + 226 h + 48805 h + 144137 h + 116) t - h (2700403 h + 173806 h 18 20 28 24 32 + 4115079 h + 2909256 h + 60260 h + 699218 h + 1549 h 14 34 36 4 2 26 + 3975690 h + 130 h + 276 + 5 h + 36625 h + 4736 h + 234794 h 16 30 22 10 8 4 + 4569659 h + 11513 h + 1613630 h + 1424276 h + 576399 h ) t + 13 14 40 32 18 22 h (15790372 h + 31 h + 165723 h + 25789913 h + 18116629 h 26 38 20 16 4 + 5412454 h + 465 h + 24036685 h + 22414273 h + 64079 h 24 12 28 36 34 + 11027671 h + 8981626 h + 2133985 h + 4524 h + 31596 h 30 42 6 8 10 2 + 670583 h + 296 + h + 366895 h + 1424867 h + 4068240 h + 6614 h ) 5 16 4 24 36 6 t - h (60346 h + 160 + 75408509 h + 270211 h + 436648 h 12 14 18 22 8 + 18575551 h + 38199266 h + 87956707 h + 95694675 h + 2084984 h 44 10 38 16 40 2 + 36 h + 7141696 h + 48252 h + 63814755 h + 6473 h + 4788 h 46 42 32 20 28 + h + 611 h + 4104843 h + 100657187 h + 26282028 h 26 34 30 6 19 14 + 49085171 h + 1180032 h + 11517241 h ) t + h (32 + 57484716 h 28 2 38 6 40 + 147603168 h + 1704 h + 1258991 h + 288511 h + 250732 h 20 30 8 22 + 245920814 h + 84037054 h + 1745368 h + 278995021 h 24 12 26 10 42 + 267626604 h + 23407408 h + 216762956 h + 7405083 h + 37433 h 4 18 34 48 44 + 30380 h + 182628324 h + 15511383 h + 8 h + 3928 h 16 46 32 36 7 24 + 113129005 h + 258 h + 39758490 h + 4935492 h ) t - h (224 14 28 2 38 6 + 123710205 h + 308299822 h + 7324 h + 3513939 h + 813811 h 40 20 30 8 22 + 741207 h + 497272904 h + 183205573 h + 4414040 h + 559189597 h 24 12 26 10 42 + 536630022 h + 52226986 h + 440323848 h + 17388604 h + 116461 h 4 18 34 48 44 + 100916 h + 374956186 h + 38070471 h + 28 h + 12787 h 16 46 32 36 8 29 + 237058972 h + 874 h + 91604846 h + 12934141 h ) t + h (608 14 28 2 38 6 + 187988547 h + 457875239 h + 16158 h + 6171064 h + 1442845 h 40 20 30 8 22 + 1344917 h + 731022920 h + 278938622 h + 7365278 h + 816623723 h 24 12 26 10 42 + 781653279 h + 81022631 h + 644427252 h + 27799270 h + 217648 h 4 18 34 48 44 + 195298 h + 555920452 h + 62145086 h + 56 h + 24536 h 16 46 32 36 9 34 + 355152952 h + 1716 h + 144115090 h + 21919574 h ) t - h (808 14 28 2 38 6 + 203474243 h + 490640575 h + 19991 h + 6990931 h + 1653437 h 40 20 30 8 22 + 1552621 h + 781140853 h + 300545982 h + 8253795 h + 871158301 h 24 12 26 10 42 + 833687486 h + 88379175 h + 688296724 h + 30661322 h + 256403 h 4 18 34 48 44 + 230898 h + 595812955 h + 68263607 h + 70 h + 29511 h 16 46 32 36 10 39 + 382248774 h + 2106 h + 156552160 h + 24419281 h ) t + h (550 14 28 2 38 6 + 156940613 h + 380359239 h + 14232 h + 5140769 h + 1230219 h 40 20 30 8 22 + 1149232 h + 606882386 h + 230000916 h + 6231676 h + 679494394 h 24 12 26 10 42 + 652000946 h + 67917979 h + 537537211 h + 23401253 h + 192375 h 4 18 34 48 44 + 168642 h + 461418505 h + 50712571 h + 56 h + 22562 h 16 46 32 36 11 44 + 295417528 h + 1646 h + 117952718 h + 17981188 h ) t - h (184 14 28 2 38 6 + 85661646 h + 211457488 h + 5627 h + 2428002 h + 589097 h 40 20 30 8 22 + 535697 h + 343177465 h + 124339354 h + 3146266 h + 386987596 h 24 12 26 10 42 + 371781801 h + 36433922 h + 304224670 h + 12248530 h + 89768 h 4 18 34 48 44 + 74727 h + 258266320 h + 25463317 h + 28 h + 10677 h 16 46 32 36 12 49 + 163429497 h + 798 h + 61539876 h + 8717101 h ) t + h (24 14 28 2 38 6 + 32716211 h + 82904016 h + 1132 h + 716186 h + 175443 h 40 20 30 8 22 + 149849 h + 140814945 h + 46749799 h + 1034673 h + 159985028 h 24 12 26 10 42 + 152966541 h + 13422287 h + 122956370 h + 4305222 h + 24307 h 4 18 34 48 44 16 + 19124 h + 104140769 h + 8552151 h + 8 h + 2873 h + 64304431 h 46 32 36 13 56 16 + 219 h + 21962569 h + 2741494 h ) t - h (29936227 h 28 8 34 2 18 + 11918125 h + 978200 h + 538854 h + 2548 h + 41436201 h 32 14 36 38 30 + 1859493 h + 17791600 h + 125039 h + 22899 h + 5203914 h 44 42 26 22 20 46 + 26 h + 92 + 350 h + 22453284 h + 44725552 h + 47361498 h + h 40 4 24 6 12 + 3258 h + 30261 h + 34880871 h + 210115 h + 8603583 h 10 14 63 32 24 18 + 3311356 h ) t + h (59789 h + 4086568 h + 9825094 h 10 20 16 40 8 + 1509473 h + 9079488 h + 8574107 h + 128 + 6 h + 522650 h 34 36 2 30 6 22 + 11334 h + 1566 h + 2628 h + 242749 h + 134917 h + 6776589 h 26 14 4 12 38 + 1988300 h + 6019737 h + 24154 h + 3384967 h + 140 h 28 15 70 20 14 12 34 + 777536 h ) t - h (868767 h + 1210224 h + 817150 h + 15 h 16 22 32 30 10 + 1388753 h + 475766 h + 303 h + 2826 h + 424151 h + 76 4 28 2 8 26 24 + 9973 h + 16385 h + 1276 h + 167110 h + 66751 h + 203133 h 18 6 16 77 2 26 24 22 + 1241909 h + 48733 h ) t + h (1 + h ) (20 h + 312 h + 2145 h 20 18 16 14 12 10 + 8737 h + 23785 h + 45784 h + 63715 h + 64373 h + 46820 h 8 6 4 2 17 86 2 18 + 24030 h + 8453 h + 1944 h + 264 h + 16) t - h (1 + h ) (15 h 16 14 12 10 8 6 4 + 173 h + 827 h + 2194 h + 3595 h + 3741 h + 2483 h + 1032 h 2 18 + 243 h + 24) t 95 2 10 8 6 4 2 19 + h (1 + h ) (6 h + 42 h + 105 h + 125 h + 67 h + 12) t 104 2 2 20 / 5 - h (2 + h ) (1 + h ) t ) / ((t h - 1) / 2 10 3 5 2 10 3 5 3 13 (t h + 1 - 2 h t - 2 t h ) (t h + 1 - h t - 2 t h ) (1 - 2 t h 3 15 4 20 2 8 3 6 2 5 2 10 - 8 t h + t h + 4 t h - 2 h t + h t - 8 t h + 6 t h 3 17 3 19 7 9 2 14 2 16 2 12 - 9 t h - 2 t h - 9 t h - 2 t h + 20 t h + 8 t h + 16 t h 2 18 4 20 3 15 3 13 3 11 2 10 2 8 + t h ) (t h - 4 t h - 6 t h - 2 t h + 6 t h + 12 t h 6 2 5 3 3 13 3 11 + 5 h t - 4 t h - 6 h t - 2 h t + 1) (1 - 722 t h - 438 t h 3 15 4 20 2 8 4 24 2 4 2 2 - 922 t h + 2074 t h + 152 t h + 1092 t h + 24 t h + 4 t h 3 6 2 7 31 5 2 10 4 12 - 20 h t + 62 h t - 4 t h - 34 t h + 320 t h + 33 t h - 4 h t 4 26 7 35 7 33 3 25 6 32 3 9 + 848 t h - 34 t h - 20 t h - 54 t h + 376 t h - 148 t h 3 7 8 40 6 34 6 26 6 24 3 17 - 20 t h + t h + 221 t h + 62 t h + 24 t h - 1178 t h 6 22 6 30 3 19 3 21 7 6 28 + 4 t h + 320 t h - 1224 t h - 796 t h - 22 t h + 152 t h 9 5 23 5 25 2 14 2 16 5 17 - 4 t h - 722 t h - 922 t h + 221 t h + 60 t h - 20 t h 3 23 2 12 4 16 5 19 2 18 - 290 t h + 376 t h + 772 t h - 148 t h + 6 t h 5 31 5 29 5 27 5 21 5 33 - 796 t h - 1224 t h - 1178 t h - 438 t h - 290 t h 5 35 7 37 5 37 4 36 4 32 4 34 - 54 t h - 22 t h - 4 t h + t h + 104 t h + 16 t h 4 30 4 22 4 14 4 28 4 18 + 352 t h + 1720 t h + 232 t h + 676 t h + 1584 t h 6 36 6 38 7 39 3 27 + 60 t h + 6 t h - 4 t h - 4 t h )) and in Maple input format: (-h^2*(1+3*h^2+h^4)*(h^4+9*h^2+9)+h^3*(441*h^12+1208*h^10+789*h^4+79*h^14+5*h^ 16+1584*h^6+1823*h^8+207*h^2+22)*t-h^4*(16+32388*h^14+21182*h^8+2060*h^4+284*h^ 2+40909*h^12+193*h^22+6479*h^18+1501*h^20+10*h^24+35668*h^10+8336*h^6+17669*h^ 16)*t^2+h^7*(2217*h^26+464091*h^16+298063*h^18+304953*h^10+10*h^30+541374*h^14+ 12909*h^24+143036*h^20+1668*h^2+471896*h^12+50714*h^22+11390*h^4+226*h^28+48805 *h^6+144137*h^8+116)*t^3-h^10*(2700403*h^12+173806*h^6+4115079*h^18+2909256*h^ 20+60260*h^28+699218*h^24+1549*h^32+3975690*h^14+130*h^34+276+5*h^36+36625*h^4+ 4736*h^2+234794*h^26+4569659*h^16+11513*h^30+1613630*h^22+1424276*h^10+576399*h ^8)*t^4+h^13*(15790372*h^14+31*h^40+165723*h^32+25789913*h^18+18116629*h^22+ 5412454*h^26+465*h^38+24036685*h^20+22414273*h^16+64079*h^4+11027671*h^24+ 8981626*h^12+2133985*h^28+4524*h^36+31596*h^34+670583*h^30+296+h^42+366895*h^6+ 1424867*h^8+4068240*h^10+6614*h^2)*t^5-h^16*(60346*h^4+160+75408509*h^24+270211 *h^36+436648*h^6+18575551*h^12+38199266*h^14+87956707*h^18+95694675*h^22+ 2084984*h^8+36*h^44+7141696*h^10+48252*h^38+63814755*h^16+6473*h^40+4788*h^2+h^ 46+611*h^42+4104843*h^32+100657187*h^20+26282028*h^28+49085171*h^26+1180032*h^ 34+11517241*h^30)*t^6+h^19*(32+57484716*h^14+147603168*h^28+1704*h^2+1258991*h^ 38+288511*h^6+250732*h^40+245920814*h^20+84037054*h^30+1745368*h^8+278995021*h^ 22+267626604*h^24+23407408*h^12+216762956*h^26+7405083*h^10+37433*h^42+30380*h^ 4+182628324*h^18+15511383*h^34+8*h^48+3928*h^44+113129005*h^16+258*h^46+ 39758490*h^32+4935492*h^36)*t^7-h^24*(224+123710205*h^14+308299822*h^28+7324*h^ 2+3513939*h^38+813811*h^6+741207*h^40+497272904*h^20+183205573*h^30+4414040*h^8 +559189597*h^22+536630022*h^24+52226986*h^12+440323848*h^26+17388604*h^10+ 116461*h^42+100916*h^4+374956186*h^18+38070471*h^34+28*h^48+12787*h^44+ 237058972*h^16+874*h^46+91604846*h^32+12934141*h^36)*t^8+h^29*(608+187988547*h^ 14+457875239*h^28+16158*h^2+6171064*h^38+1442845*h^6+1344917*h^40+731022920*h^ 20+278938622*h^30+7365278*h^8+816623723*h^22+781653279*h^24+81022631*h^12+ 644427252*h^26+27799270*h^10+217648*h^42+195298*h^4+555920452*h^18+62145086*h^ 34+56*h^48+24536*h^44+355152952*h^16+1716*h^46+144115090*h^32+21919574*h^36)*t^ 9-h^34*(808+203474243*h^14+490640575*h^28+19991*h^2+6990931*h^38+1653437*h^6+ 1552621*h^40+781140853*h^20+300545982*h^30+8253795*h^8+871158301*h^22+833687486 *h^24+88379175*h^12+688296724*h^26+30661322*h^10+256403*h^42+230898*h^4+ 595812955*h^18+68263607*h^34+70*h^48+29511*h^44+382248774*h^16+2106*h^46+ 156552160*h^32+24419281*h^36)*t^10+h^39*(550+156940613*h^14+380359239*h^28+ 14232*h^2+5140769*h^38+1230219*h^6+1149232*h^40+606882386*h^20+230000916*h^30+ 6231676*h^8+679494394*h^22+652000946*h^24+67917979*h^12+537537211*h^26+23401253 *h^10+192375*h^42+168642*h^4+461418505*h^18+50712571*h^34+56*h^48+22562*h^44+ 295417528*h^16+1646*h^46+117952718*h^32+17981188*h^36)*t^11-h^44*(184+85661646* h^14+211457488*h^28+5627*h^2+2428002*h^38+589097*h^6+535697*h^40+343177465*h^20 +124339354*h^30+3146266*h^8+386987596*h^22+371781801*h^24+36433922*h^12+ 304224670*h^26+12248530*h^10+89768*h^42+74727*h^4+258266320*h^18+25463317*h^34+ 28*h^48+10677*h^44+163429497*h^16+798*h^46+61539876*h^32+8717101*h^36)*t^12+h^ 49*(24+32716211*h^14+82904016*h^28+1132*h^2+716186*h^38+175443*h^6+149849*h^40+ 140814945*h^20+46749799*h^30+1034673*h^8+159985028*h^22+152966541*h^24+13422287 *h^12+122956370*h^26+4305222*h^10+24307*h^42+19124*h^4+104140769*h^18+8552151*h ^34+8*h^48+2873*h^44+64304431*h^16+219*h^46+21962569*h^32+2741494*h^36)*t^13-h^ 56*(29936227*h^16+11918125*h^28+978200*h^8+538854*h^34+2548*h^2+41436201*h^18+ 1859493*h^32+17791600*h^14+125039*h^36+22899*h^38+5203914*h^30+26*h^44+92+350*h ^42+22453284*h^26+44725552*h^22+47361498*h^20+h^46+3258*h^40+30261*h^4+34880871 *h^24+210115*h^6+8603583*h^12+3311356*h^10)*t^14+h^63*(59789*h^32+4086568*h^24+ 9825094*h^18+1509473*h^10+9079488*h^20+8574107*h^16+128+6*h^40+522650*h^8+11334 *h^34+1566*h^36+2628*h^2+242749*h^30+134917*h^6+6776589*h^22+1988300*h^26+ 6019737*h^14+24154*h^4+3384967*h^12+140*h^38+777536*h^28)*t^15-h^70*(868767*h^ 20+1210224*h^14+817150*h^12+15*h^34+1388753*h^16+475766*h^22+303*h^32+2826*h^30 +424151*h^10+76+9973*h^4+16385*h^28+1276*h^2+167110*h^8+66751*h^26+203133*h^24+ 1241909*h^18+48733*h^6)*t^16+h^77*(1+h^2)*(20*h^26+312*h^24+2145*h^22+8737*h^20 +23785*h^18+45784*h^16+63715*h^14+64373*h^12+46820*h^10+24030*h^8+8453*h^6+1944 *h^4+264*h^2+16)*t^17-h^86*(1+h^2)*(15*h^18+173*h^16+827*h^14+2194*h^12+3595*h^ 10+3741*h^8+2483*h^6+1032*h^4+243*h^2+24)*t^18+h^95*(1+h^2)*(6*h^10+42*h^8+105* h^6+125*h^4+67*h^2+12)*t^19-h^104*(2+h^2)*(1+h^2)*t^20)/(t*h^5-1)/(t^2*h^10+1-2 *h^3*t-2*t*h^5)/(t^2*h^10+1-h^3*t-2*t*h^5)/(1-2*t^3*h^13-8*t^3*h^15+t^4*h^20+4* t^2*h^8-2*h^3*t+h^6*t^2-8*t*h^5+6*t^2*h^10-9*t^3*h^17-2*t^3*h^19-9*t*h^7-2*t*h^ 9+20*t^2*h^14+8*t^2*h^16+16*t^2*h^12+t^2*h^18)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-\ 2*t^3*h^11+6*t^2*h^10+12*t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1)/(1-722*t^3* h^13-438*t^3*h^11-922*t^3*h^15+2074*t^4*h^20+152*t^2*h^8+1092*t^4*h^24+24*t^2*h ^4+4*t^2*h^2-20*h^3*t+62*h^6*t^2-4*t^7*h^31-34*t*h^5+320*t^2*h^10+33*t^4*h^12-4 *h*t+848*t^4*h^26-34*t^7*h^35-20*t^7*h^33-54*t^3*h^25+376*t^6*h^32-148*t^3*h^9-\ 20*t^3*h^7+t^8*h^40+221*t^6*h^34+62*t^6*h^26+24*t^6*h^24-1178*t^3*h^17+4*t^6*h^ 22+320*t^6*h^30-1224*t^3*h^19-796*t^3*h^21-22*t*h^7+152*t^6*h^28-4*t*h^9-722*t^ 5*h^23-922*t^5*h^25+221*t^2*h^14+60*t^2*h^16-20*t^5*h^17-290*t^3*h^23+376*t^2*h ^12+772*t^4*h^16-148*t^5*h^19+6*t^2*h^18-796*t^5*h^31-1224*t^5*h^29-1178*t^5*h^ 27-438*t^5*h^21-290*t^5*h^33-54*t^5*h^35-22*t^7*h^37-4*t^5*h^37+t^4*h^36+104*t^ 4*h^32+16*t^4*h^34+352*t^4*h^30+1720*t^4*h^22+232*t^4*h^14+676*t^4*h^28+1584*t^ 4*h^18+60*t^6*h^36+6*t^6*h^38-4*t^7*h^39-4*t^3*h^27) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 72 _Z + 338 _Z - 72 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 72 b + 338 b - 72 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0149322967114543 We have the following proven facts The total number of tilings of the region is asymptotic to: /4215091 21121393 4511863 2 62677 3\ n |------- - -------- b + ------- b - ----- b | (1/b) \ 46200 46200 46200 46200 / that in floating-point is: n 84.4308638498834 66.9689344729479 and in Maple input format: HFloat(84.4308638498833801)*HFloat(66.9689344729478790)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 14527 46573 b 52067 b 92 b /21031 1941 2665 2 31 3\ ----- + ------- - -------- + ----- + |----- + ---- b - ---- b + --- b | n 3080 2310 9240 1155 \4620 154 924 770 / and in Maple-input format: 14527/3080+46573/2310*b-52067/9240*b^2+92/1155*b^3+(21031/4620+1941/154*b-2665/ 924*b^2+31/770*b^3)*n and in floating point: 5.01635934124667 + 4.73972665245616 n The asymptotic expression for the variance, as a function of n is: 3 2 4393 1219 b 406831 b 93529 b /4789 292 521 2 3\ ---- + ------- - --------- - ------- + |---- - --- b + --- b - 4/175 b | n 1386 6930 34650 34650 \1575 35 315 / and in Maple-input format: 4393/1386+1219/6930*b^3-406831/34650*b^2-93529/34650*b+(4789/1575-292/35*b+521/ 315*b^2-4/175*b^3)*n and in floating point: 3.12662929303916 + 2.91642561768438 n The even alpha coefficients until the, 8, -th , are: 10071977 1860345 2663935 2 1359 3 - -------- - ------- b + -------- b - ------- b 31565996 7891499 31565996 1127357 / [1, 3 + ------------------------------------------------- + | n \ 4229721857689 31087513402069 282029554414063 2 2098744183369 3\ - ------------- + -------------- b - --------------- b + ------------- b | 267363986120 26736398612 53472797224 26736398612 / / 2 / n , 15 / 127691880753 13381073505 24298021815 2 12501351 3 - ------------ - ----------- b + ----------- b - --------- b 26736398612 6684099653 26736398612 954871379 / + -------------------------------------------------------------- + | n \ 10811342765266389 198721211446211427 3611459671931089383 2 - ----------------- + ------------------ b - ------------------- b 45291459248728 11322864812182 45291459248728 13437546113629557 3\ / 2 + ----------------- b | / n , 105 11322864812182 / / 127419293721 6252927570 14750749455 2 15473214 3 - ------------ - ---------- b + ----------- b - --------- b 1909742758 954871379 1909742758 136410197 / + ------------------------------------------------------------- + | n \ 10861484120411385 199919450902128189 3639700061518595631 2 - ----------------- + ------------------ b - ------------------- b 3235104232052 808776058013 3235104232052 13542702463821039 3\ / 2 + ----------------- b | / n ] 808776058013 / / and in Maple-input format: [1, 3+(-10071977/31565996-1860345/7891499*b+2663935/31565996*b^2-1359/1127357*b ^3)/n+(-4229721857689/267363986120+31087513402069/26736398612*b-282029554414063 /53472797224*b^2+2098744183369/26736398612*b^3)/n^2, 15+(-127691880753/ 26736398612-13381073505/6684099653*b+24298021815/26736398612*b^2-12501351/ 954871379*b^3)/n+(-10811342765266389/45291459248728+198721211446211427/ 11322864812182*b-3611459671931089383/45291459248728*b^2+13437546113629557/ 11322864812182*b^3)/n^2, 105+(-127419293721/1909742758-6252927570/954871379*b+ 14750749455/1909742758*b^2-15473214/136410197*b^3)/n+(-10861484120411385/ 3235104232052+199919450902128189/808776058013*b-3639700061518595631/ 3235104232052*b^2+13542702463821039/808776058013*b^3)/n^2] and in floating-point it is: 0.322578132263974 0.366549734211393 [1, 3 - ----------------- + -----------------, n 2 n 4.80564740984938 5.58676672715327 15 - ---------------- + ----------------, n 2 n 66.8167197023482 82.8924981567179 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 34073379 1025460972 238681809 2 2382372 3 ----------- + ---------- b - ---------- b + ---------- b 33420498265 6684099653 6684099653 4774356895 / 10244587213269 [---------------------------------------------------------- + |- -------------- n \ 56614324060910 346580913584418 3550140778981761 2 26428615884978 3\ / 2 + --------------- b - ---------------- b + -------------- b | / n , 28307162030455 56614324060910 28307162030455 / / 681467580 102546097200 23868180900 2 47647440 3 ---------- + ------------ b - ----------- b + --------- b 6684099653 6684099653 6684099653 954871379 / ----------------------------------------------------------- + | n \ 101221026247350 6821684616903285 5068274354397630 2 - --------------- + ---------------- b - ---------------- b 5661432406091 5661432406091 808776058013 528246105074085 3\ / 2 + --------------- b | / n , 5661432406091 / / 1533302055 230728718700 53703407025 2 107206740 3 ---------- + ------------ b - ----------- b + --------- b 136410197 136410197 136410197 19487171 / ----------------------------------------------------------- + | n \ 4954024363336695 2318243311667394765 12283384700649461265 2 - ---------------- + ------------------- b - -------------------- b 2541867610898 17793073276286 17793073276286 182902348381198365 3\ / 2 + ------------------ b | / n ] 17793073276286 / / and in Maple-input format: [(34073379/33420498265+1025460972/6684099653*b-238681809/6684099653*b^2+2382372 /4774356895*b^3)/n+(-10244587213269/56614324060910+346580913584418/ 28307162030455*b-3550140778981761/56614324060910*b^2+26428615884978/ 28307162030455*b^3)/n^2, (681467580/6684099653+102546097200/6684099653*b-\ 23868180900/6684099653*b^2+47647440/954871379*b^3)/n+(-101221026247350/ 5661432406091+6821684616903285/5661432406091*b-5068274354397630/808776058013*b^ 2+528246105074085/5661432406091*b^3)/n^2, (1533302055/136410197+230728718700/ 136410197*b-53703407025/136410197*b^2+107206740/19487171*b^3)/n+(-\ 4954024363336695/2541867610898+2318243311667394765/17793073276286*b-\ 12283384700649461265/17793073276286*b^2+182902348381198365/17793073276286*b^3)/ n^2] and in floating-point it is: 0.00330245744902289 0.0121082504372843 0.330245744902289 1.28350631922617 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 36.4095933740273 157.349640550966 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 35, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 3, d[2] = 1 Then infinity ----- \ n 4 2 2 2 ) A(n, h) t = (-(h + 7 h + 1) (1 + h ) / ----- n = 0 2 2 12 10 8 6 4 2 4 + h (1 + h ) (2 h + 21 h + 73 h + 109 h + 90 h + 36 h + 4) t - h 4 2 6 16 14 20 18 8 (207 h + 4 + 44 h + 584 h + 96 h + 345 h + h + 15 h + 1027 h 10 12 2 8 8 10 14 20 + 1122 h + 774 h ) t + h (2021 h + 2504 h + 1381 h + 18 h 2 22 6 4 16 12 18 3 + 136 h + h + 1228 h + 526 h + 549 h + 2255 h + 16 + 134 h ) t 12 8 12 10 20 4 18 - h (2236 h + 2580 h + 20 + 2776 h + 32 h + 583 h + 212 h 22 2 16 6 14 4 16 12 + 2 h + 160 h + 770 h + 1348 h + 1724 h ) t + h (1307 h 14 22 4 2 6 8 10 20 + 735 h + h + 296 h + 68 h + 798 h + 1397 h + 1634 h + 14 h 18 16 5 22 + 84 h + 8 + 299 h ) t - h ( 14 4 16 10 12 6 2 8 32 h + 222 h + 3 h + 305 h + 135 h + 383 h + 77 h + 12 + 426 h ) 6 28 2 4 2 4 2 7 34 2 2 8 t + h (1 + h ) (3 h + 5 h + 3) (h + 4 h + 2) t - h (1 + h ) t ) / 4 2 8 4 6 8 / ((t h - 1) (t h + 1 - 2 t h - 4 t h - t h ) / 2 8 2 4 4 16 3 16 2 16 3 14 (t h + 1 - 2 h t - 2 t h ) (t h - 2 t h + t h - 10 t h 2 14 3 12 2 12 3 10 2 10 2 8 + 8 t h - 12 t h + 20 t h - 4 t h + 16 t h + 6 t h 8 6 2 6 2 4 4 2 - 2 t h + 8 h t - 10 t h + 4 t h - 12 t h - 4 h t + 1)) and in Maple input format: (-(h^4+7*h^2+1)*(1+h^2)^2+h^2*(1+h^2)*(2*h^12+21*h^10+73*h^8+109*h^6+90*h^4+36* h^2+4)*t-h^4*(207*h^4+4+44*h^2+584*h^6+96*h^16+345*h^14+h^20+15*h^18+1027*h^8+ 1122*h^10+774*h^12)*t^2+h^8*(2021*h^8+2504*h^10+1381*h^14+18*h^20+136*h^2+h^22+ 1228*h^6+526*h^4+549*h^16+2255*h^12+16+134*h^18)*t^3-h^12*(2236*h^8+2580*h^12+ 20+2776*h^10+32*h^20+583*h^4+212*h^18+2*h^22+160*h^2+770*h^16+1348*h^6+1724*h^ 14)*t^4+h^16*(1307*h^12+735*h^14+h^22+296*h^4+68*h^2+798*h^6+1397*h^8+1634*h^10 +14*h^20+84*h^18+8+299*h^16)*t^5-h^22*(32*h^14+222*h^4+3*h^16+305*h^10+135*h^12 +383*h^6+77*h^2+12+426*h^8)*t^6+h^28*(1+h^2)*(3*h^4+5*h^2+3)*(h^4+4*h^2+2)*t^7- h^34*(1+h^2)^2*t^8)/(t*h^4-1)/(t^2*h^8+1-2*t*h^4-4*t*h^6-t*h^8)/(t^2*h^8+1-2*h^ 2*t-2*t*h^4)/(t^4*h^16-2*t^3*h^16+t^2*h^16-10*t^3*h^14+8*t^2*h^14-12*t^3*h^12+ 20*t^2*h^12-4*t^3*h^10+16*t^2*h^10+6*t^2*h^8-2*t*h^8+8*h^6*t^2-10*t*h^6+4*t^2*h ^4-12*t*h^4-4*h^2*t+1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 28 _Z + 63 _Z - 28 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 28 b + 63 b - 28 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0390932603206857 We have the following proven facts The total number of tilings of the region is asymptotic to: /63647 17041 7687 2 1237 3\ n |----- - ----- b + ---- b - ---- b | (1/b) \1980 220 220 990 / that in floating-point is: n 29.1701460305608 25.5798567783016 and in Maple input format: HFloat(29.1701460305607725)*HFloat(25.5798567783015649)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 1721 8996 b 9484 b 119 b /716 256 184 2 34 3\ ---- + ------ - ------- + ------ + |--- + --- b - --- b + --- b | n 495 495 495 165 \165 33 33 165 / and in Maple-input format: 1721/495+8996/495*b-9484/495*b^2+119/165*b^3+(716/165+256/33*b-184/33*b^2+34/ 165*b^3)*n and in floating point: 4.15800012753731 + 4.63415384322664 n The asymptotic expression for the variance, as a function of n is: 3 2 1456 11698 b 917648 b 509272 b ---- + -------- - --------- + -------- 7425 2475 7425 7425 / 1856 608 2 316 3 17416\ + |- ---- b + --- b - ---- b + -----| n \ 495 495 7425 7425 / and in Maple-input format: 1456/7425+11698/2475*b^3-917648/7425*b^2+509272/7425*b+(-1856/495*b+608/495*b^2 -316/7425*b^3+17416/7425)*n and in floating point: 2.68885841619758 + 2.20088386493014 n The even alpha coefficients until the, 8, -th , are: 1434034 338320 44840 2 6641 3 - ------- + ------- b + ------- b - ------- b 3110217 1036739 1036739 3110217 / 21969685269856 [1, 3 + ----------------------------------------------- + |- -------------- n \ 4774183095 120652827161038 211191088530932 2 5381346012815 3\ / 2 + --------------- b - --------------- b + ------------- b | / n , 15 954836619 954836619 636557746 / / 230378218 1170459120 676704360 2 24690643 3 - --------- + ---------- b - --------- b + -------- b 28934443 28934443 28934443 28934443 / + ------------------------------------------------------- + | n \ 6957636763169604 191048766088465162 334457385536666108 2 - ---------------- + ------------------ b - ------------------ b 97711614011 97711614011 97711614011 25566937383188123 3\ / 2 + ----------------- b | / n , 105 195423228022 / / 40139100484 338689918560 211315777680 2 7747433134 3 - ----------- + ------------ b - ------------ b + ---------- b 318278873 318278873 318278873 318278873 / + ---------------------------------------------------------------- + | n \ 100381232738332861 2756365931861602956 4826233409563180104 2 - ------------------ + ------------------- b - ------------------- b 97711614011 97711614011 97711614011 184466688465337117 3\ / 2 + ------------------ b | / n ] 97711614011 / / and in Maple-input format: [1, 3+(-1434034/3110217+338320/1036739*b+44840/1036739*b^2-6641/3110217*b^3)/n+ (-21969685269856/4774183095+120652827161038/954836619*b-211191088530932/ 954836619*b^2+5381346012815/636557746*b^3)/n^2, 15+(-230378218/28934443+ 1170459120/28934443*b-676704360/28934443*b^2+24690643/28934443*b^3)/n+(-\ 6957636763169604/97711614011+191048766088465162/97711614011*b-\ 334457385536666108/97711614011*b^2+25566937383188123/195423228022*b^3)/n^2, 105 +(-40139100484/318278873+338689918560/318278873*b-211315777680/318278873*b^2+ 7747433134/318278873*b^3)/n+(-100381232738332861/97711614011+ 2756365931861602956/97711614011*b-4826233409563180104/97711614011*b^2+ 184466688465337117/97711614011*b^3)/n^2] and in floating-point it is: 0.448248696679244 0.521390865623671 [1, 3 - ----------------- + -----------------, n 2 n 6.41636251823189 7.17009859652749 15 - ---------------- + ----------------, n 2 n 85.5259242078518 94.8577919234394 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 166459104 1131708672 765023616 2 140895504 3 - ---------- + ---------- b - --------- b + ---------- b 1591394365 318278873 318278873 1591394365 / 9690578062688 [---------------------------------------------------------- + |- ------------- n \ 44414370005 265995727357136 467337563593024 2 17863811162032 3\ / 2 + --------------- b - --------------- b + -------------- b | / n , 44414370005 44414370005 44414370005 / / 3329182080 113170867200 76502361600 2 2817910080 3 - ---------- + ------------ b - ----------- b + ---------- b 318278873 318278873 318278873 318278873 / -------------------------------------------------------------- + | n \ 193611642474400 5314424312751520 9344271811167680 2 - --------------- + ---------------- b - ---------------- b 8882874001 8882874001 8882874001 357187671073040 3\ / 2 + --------------- b | / n , 8882874001 / / 367042324320 12477088108800 8434385366400 2 310674586320 3 - ------------ + -------------- b - ------------- b + ------------ b 318278873 318278873 318278873 318278873 / ---------------------------------------------------------------------- + | n \ 6546032251382902320 179674484780934480960 316182229067425724640 2 - ------------------- + --------------------- b - --------------------- b 2727042318307 2727042318307 2727042318307 12086391703848061320 3\ / 2 + -------------------- b | / n ] 2727042318307 / / and in Maple-input format: [(-166459104/1591394365+1131708672/318278873*b-765023616/318278873*b^2+ 140895504/1591394365*b^3)/n+(-9690578062688/44414370005+265995727357136/ 44414370005*b-467337563593024/44414370005*b^2+17863811162032/44414370005*b^3)/n ^2, (-3329182080/318278873+113170867200/318278873*b-76502361600/318278873*b^2+ 2817910080/318278873*b^3)/n+(-193611642474400/8882874001+5314424312751520/ 8882874001*b-9344271811167680/8882874001*b^2+357187671073040/8882874001*b^3)/n^ 2, (-367042324320/318278873+12477088108800/318278873*b-8434385366400/318278873* b^2+310674586320/318278873*b^3)/n+(-6546032251382902320/2727042318307+ 179674484780934480960/2727042318307*b-316182229067425724640/2727042318307*b^2+ 12086391703848061320/2727042318307*b^3)/n^2] and in floating-point it is: 0.0307367931956769 0.114706617065132 3.07367931956768 12.7009262232392 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 338.873145154837 1637.18096304436 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 36, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 3, d[2] = 2 Then infinity ----- \ n 2 2 4 4 2 3 12 ) A(n, h) t = (-h (1 + 3 h + h ) (h + 9 h + 9) + h (441 h / ----- n = 0 10 4 14 16 6 8 2 + 1208 h + 789 h + 79 h + 5 h + 1584 h + 1823 h + 207 h + 22) t 4 14 8 4 2 12 22 - h (16 + 32388 h + 21182 h + 2060 h + 284 h + 40909 h + 193 h 18 20 24 10 6 16 2 7 + 6479 h + 1501 h + 10 h + 35668 h + 8336 h + 17669 h ) t + h 26 16 18 10 30 14 (2217 h + 464091 h + 298063 h + 304953 h + 10 h + 541374 h 24 20 2 12 22 4 + 12909 h + 143036 h + 1668 h + 471896 h + 50714 h + 11390 h 28 6 8 3 10 12 6 + 226 h + 48805 h + 144137 h + 116) t - h (2700403 h + 173806 h 18 20 28 24 32 + 4115079 h + 2909256 h + 60260 h + 699218 h + 1549 h 14 34 36 4 2 26 + 3975690 h + 130 h + 276 + 5 h + 36625 h + 4736 h + 234794 h 16 30 22 10 8 4 + 4569659 h + 11513 h + 1613630 h + 1424276 h + 576399 h ) t + 13 14 40 32 18 22 h (15790372 h + 31 h + 165723 h + 25789913 h + 18116629 h 26 38 20 16 4 + 5412454 h + 465 h + 24036685 h + 22414273 h + 64079 h 24 12 28 36 34 + 11027671 h + 8981626 h + 2133985 h + 4524 h + 31596 h 30 42 6 8 10 2 + 670583 h + 296 + h + 366895 h + 1424867 h + 4068240 h + 6614 h ) 5 16 4 24 36 6 t - h (60346 h + 160 + 75408509 h + 270211 h + 436648 h 12 14 18 22 8 + 18575551 h + 38199266 h + 87956707 h + 95694675 h + 2084984 h 44 10 38 16 40 2 + 36 h + 7141696 h + 48252 h + 63814755 h + 6473 h + 4788 h 46 42 32 20 28 + h + 611 h + 4104843 h + 100657187 h + 26282028 h 26 34 30 6 19 14 + 49085171 h + 1180032 h + 11517241 h ) t + h (32 + 57484716 h 28 2 38 6 40 + 147603168 h + 1704 h + 1258991 h + 288511 h + 250732 h 20 30 8 22 + 245920814 h + 84037054 h + 1745368 h + 278995021 h 24 12 26 10 42 + 267626604 h + 23407408 h + 216762956 h + 7405083 h + 37433 h 4 18 34 48 44 + 30380 h + 182628324 h + 15511383 h + 8 h + 3928 h 16 46 32 36 7 24 + 113129005 h + 258 h + 39758490 h + 4935492 h ) t - h (224 14 28 2 38 6 + 123710205 h + 308299822 h + 7324 h + 3513939 h + 813811 h 40 20 30 8 22 + 741207 h + 497272904 h + 183205573 h + 4414040 h + 559189597 h 24 12 26 10 42 + 536630022 h + 52226986 h + 440323848 h + 17388604 h + 116461 h 4 18 34 48 44 + 100916 h + 374956186 h + 38070471 h + 28 h + 12787 h 16 46 32 36 8 29 + 237058972 h + 874 h + 91604846 h + 12934141 h ) t + h (608 14 28 2 38 6 + 187988547 h + 457875239 h + 16158 h + 6171064 h + 1442845 h 40 20 30 8 22 + 1344917 h + 731022920 h + 278938622 h + 7365278 h + 816623723 h 24 12 26 10 42 + 781653279 h + 81022631 h + 644427252 h + 27799270 h + 217648 h 4 18 34 48 44 + 195298 h + 555920452 h + 62145086 h + 56 h + 24536 h 16 46 32 36 9 34 + 355152952 h + 1716 h + 144115090 h + 21919574 h ) t - h (808 14 28 2 38 6 + 203474243 h + 490640575 h + 19991 h + 6990931 h + 1653437 h 40 20 30 8 22 + 1552621 h + 781140853 h + 300545982 h + 8253795 h + 871158301 h 24 12 26 10 42 + 833687486 h + 88379175 h + 688296724 h + 30661322 h + 256403 h 4 18 34 48 44 + 230898 h + 595812955 h + 68263607 h + 70 h + 29511 h 16 46 32 36 10 39 + 382248774 h + 2106 h + 156552160 h + 24419281 h ) t + h (550 14 28 2 38 6 + 156940613 h + 380359239 h + 14232 h + 5140769 h + 1230219 h 40 20 30 8 22 + 1149232 h + 606882386 h + 230000916 h + 6231676 h + 679494394 h 24 12 26 10 42 + 652000946 h + 67917979 h + 537537211 h + 23401253 h + 192375 h 4 18 34 48 44 + 168642 h + 461418505 h + 50712571 h + 56 h + 22562 h 16 46 32 36 11 44 + 295417528 h + 1646 h + 117952718 h + 17981188 h ) t - h (184 14 28 2 38 6 + 85661646 h + 211457488 h + 5627 h + 2428002 h + 589097 h 40 20 30 8 22 + 535697 h + 343177465 h + 124339354 h + 3146266 h + 386987596 h 24 12 26 10 42 + 371781801 h + 36433922 h + 304224670 h + 12248530 h + 89768 h 4 18 34 48 44 + 74727 h + 258266320 h + 25463317 h + 28 h + 10677 h 16 46 32 36 12 49 + 163429497 h + 798 h + 61539876 h + 8717101 h ) t + h (24 14 28 2 38 6 + 32716211 h + 82904016 h + 1132 h + 716186 h + 175443 h 40 20 30 8 22 + 149849 h + 140814945 h + 46749799 h + 1034673 h + 159985028 h 24 12 26 10 42 + 152966541 h + 13422287 h + 122956370 h + 4305222 h + 24307 h 4 18 34 48 44 16 + 19124 h + 104140769 h + 8552151 h + 8 h + 2873 h + 64304431 h 46 32 36 13 56 16 + 219 h + 21962569 h + 2741494 h ) t - h (29936227 h 28 8 34 2 18 + 11918125 h + 978200 h + 538854 h + 2548 h + 41436201 h 32 14 36 38 30 + 1859493 h + 17791600 h + 125039 h + 22899 h + 5203914 h 44 42 26 22 20 46 + 26 h + 92 + 350 h + 22453284 h + 44725552 h + 47361498 h + h 40 4 24 6 12 + 3258 h + 30261 h + 34880871 h + 210115 h + 8603583 h 10 14 63 32 24 18 + 3311356 h ) t + h (59789 h + 4086568 h + 9825094 h 10 20 16 40 8 + 1509473 h + 9079488 h + 8574107 h + 128 + 6 h + 522650 h 34 36 2 30 6 22 + 11334 h + 1566 h + 2628 h + 242749 h + 134917 h + 6776589 h 26 14 4 12 38 + 1988300 h + 6019737 h + 24154 h + 3384967 h + 140 h 28 15 70 20 14 12 34 + 777536 h ) t - h (868767 h + 1210224 h + 817150 h + 15 h 16 22 32 30 10 + 1388753 h + 475766 h + 303 h + 2826 h + 424151 h + 76 4 28 2 8 26 24 + 9973 h + 16385 h + 1276 h + 167110 h + 66751 h + 203133 h 18 6 16 77 2 26 24 22 + 1241909 h + 48733 h ) t + h (1 + h ) (20 h + 312 h + 2145 h 20 18 16 14 12 10 + 8737 h + 23785 h + 45784 h + 63715 h + 64373 h + 46820 h 8 6 4 2 17 86 2 18 + 24030 h + 8453 h + 1944 h + 264 h + 16) t - h (1 + h ) (15 h 16 14 12 10 8 6 4 + 173 h + 827 h + 2194 h + 3595 h + 3741 h + 2483 h + 1032 h 2 18 + 243 h + 24) t 95 2 10 8 6 4 2 19 + h (1 + h ) (6 h + 42 h + 105 h + 125 h + 67 h + 12) t 104 2 2 20 / 5 - h (2 + h ) (1 + h ) t ) / ((t h - 1) / 2 10 3 5 2 10 3 5 3 13 (t h + 1 - 2 h t - 2 t h ) (t h + 1 - h t - 2 t h ) (1 - 2 t h 3 15 4 20 2 8 3 6 2 5 2 10 - 8 t h + t h + 4 t h - 2 h t + h t - 8 t h + 6 t h 3 17 3 19 7 9 2 14 2 16 2 12 - 9 t h - 2 t h - 9 t h - 2 t h + 20 t h + 8 t h + 16 t h 2 18 4 20 3 15 3 13 3 11 2 10 2 8 + t h ) (t h - 4 t h - 6 t h - 2 t h + 6 t h + 12 t h 6 2 5 3 3 13 3 11 + 5 h t - 4 t h - 6 h t - 2 h t + 1) (1 - 722 t h - 438 t h 3 15 4 20 2 8 4 24 2 4 2 2 - 922 t h + 2074 t h + 152 t h + 1092 t h + 24 t h + 4 t h 3 6 2 7 31 5 2 10 4 12 - 20 h t + 62 h t - 4 t h - 34 t h + 320 t h + 33 t h - 4 h t 4 26 7 35 7 33 3 25 6 32 3 9 + 848 t h - 34 t h - 20 t h - 54 t h + 376 t h - 148 t h 3 7 8 40 6 34 6 26 6 24 3 17 - 20 t h + t h + 221 t h + 62 t h + 24 t h - 1178 t h 6 22 6 30 3 19 3 21 7 6 28 + 4 t h + 320 t h - 1224 t h - 796 t h - 22 t h + 152 t h 9 5 23 5 25 2 14 2 16 5 17 - 4 t h - 722 t h - 922 t h + 221 t h + 60 t h - 20 t h 3 23 2 12 4 16 5 19 2 18 - 290 t h + 376 t h + 772 t h - 148 t h + 6 t h 5 31 5 29 5 27 5 21 5 33 - 796 t h - 1224 t h - 1178 t h - 438 t h - 290 t h 5 35 7 37 5 37 4 36 4 32 4 34 - 54 t h - 22 t h - 4 t h + t h + 104 t h + 16 t h 4 30 4 22 4 14 4 28 4 18 + 352 t h + 1720 t h + 232 t h + 676 t h + 1584 t h 6 36 6 38 7 39 3 27 + 60 t h + 6 t h - 4 t h - 4 t h )) and in Maple input format: (-h^2*(1+3*h^2+h^4)*(h^4+9*h^2+9)+h^3*(441*h^12+1208*h^10+789*h^4+79*h^14+5*h^ 16+1584*h^6+1823*h^8+207*h^2+22)*t-h^4*(16+32388*h^14+21182*h^8+2060*h^4+284*h^ 2+40909*h^12+193*h^22+6479*h^18+1501*h^20+10*h^24+35668*h^10+8336*h^6+17669*h^ 16)*t^2+h^7*(2217*h^26+464091*h^16+298063*h^18+304953*h^10+10*h^30+541374*h^14+ 12909*h^24+143036*h^20+1668*h^2+471896*h^12+50714*h^22+11390*h^4+226*h^28+48805 *h^6+144137*h^8+116)*t^3-h^10*(2700403*h^12+173806*h^6+4115079*h^18+2909256*h^ 20+60260*h^28+699218*h^24+1549*h^32+3975690*h^14+130*h^34+276+5*h^36+36625*h^4+ 4736*h^2+234794*h^26+4569659*h^16+11513*h^30+1613630*h^22+1424276*h^10+576399*h ^8)*t^4+h^13*(15790372*h^14+31*h^40+165723*h^32+25789913*h^18+18116629*h^22+ 5412454*h^26+465*h^38+24036685*h^20+22414273*h^16+64079*h^4+11027671*h^24+ 8981626*h^12+2133985*h^28+4524*h^36+31596*h^34+670583*h^30+296+h^42+366895*h^6+ 1424867*h^8+4068240*h^10+6614*h^2)*t^5-h^16*(60346*h^4+160+75408509*h^24+270211 *h^36+436648*h^6+18575551*h^12+38199266*h^14+87956707*h^18+95694675*h^22+ 2084984*h^8+36*h^44+7141696*h^10+48252*h^38+63814755*h^16+6473*h^40+4788*h^2+h^ 46+611*h^42+4104843*h^32+100657187*h^20+26282028*h^28+49085171*h^26+1180032*h^ 34+11517241*h^30)*t^6+h^19*(32+57484716*h^14+147603168*h^28+1704*h^2+1258991*h^ 38+288511*h^6+250732*h^40+245920814*h^20+84037054*h^30+1745368*h^8+278995021*h^ 22+267626604*h^24+23407408*h^12+216762956*h^26+7405083*h^10+37433*h^42+30380*h^ 4+182628324*h^18+15511383*h^34+8*h^48+3928*h^44+113129005*h^16+258*h^46+ 39758490*h^32+4935492*h^36)*t^7-h^24*(224+123710205*h^14+308299822*h^28+7324*h^ 2+3513939*h^38+813811*h^6+741207*h^40+497272904*h^20+183205573*h^30+4414040*h^8 +559189597*h^22+536630022*h^24+52226986*h^12+440323848*h^26+17388604*h^10+ 116461*h^42+100916*h^4+374956186*h^18+38070471*h^34+28*h^48+12787*h^44+ 237058972*h^16+874*h^46+91604846*h^32+12934141*h^36)*t^8+h^29*(608+187988547*h^ 14+457875239*h^28+16158*h^2+6171064*h^38+1442845*h^6+1344917*h^40+731022920*h^ 20+278938622*h^30+7365278*h^8+816623723*h^22+781653279*h^24+81022631*h^12+ 644427252*h^26+27799270*h^10+217648*h^42+195298*h^4+555920452*h^18+62145086*h^ 34+56*h^48+24536*h^44+355152952*h^16+1716*h^46+144115090*h^32+21919574*h^36)*t^ 9-h^34*(808+203474243*h^14+490640575*h^28+19991*h^2+6990931*h^38+1653437*h^6+ 1552621*h^40+781140853*h^20+300545982*h^30+8253795*h^8+871158301*h^22+833687486 *h^24+88379175*h^12+688296724*h^26+30661322*h^10+256403*h^42+230898*h^4+ 595812955*h^18+68263607*h^34+70*h^48+29511*h^44+382248774*h^16+2106*h^46+ 156552160*h^32+24419281*h^36)*t^10+h^39*(550+156940613*h^14+380359239*h^28+ 14232*h^2+5140769*h^38+1230219*h^6+1149232*h^40+606882386*h^20+230000916*h^30+ 6231676*h^8+679494394*h^22+652000946*h^24+67917979*h^12+537537211*h^26+23401253 *h^10+192375*h^42+168642*h^4+461418505*h^18+50712571*h^34+56*h^48+22562*h^44+ 295417528*h^16+1646*h^46+117952718*h^32+17981188*h^36)*t^11-h^44*(184+85661646* h^14+211457488*h^28+5627*h^2+2428002*h^38+589097*h^6+535697*h^40+343177465*h^20 +124339354*h^30+3146266*h^8+386987596*h^22+371781801*h^24+36433922*h^12+ 304224670*h^26+12248530*h^10+89768*h^42+74727*h^4+258266320*h^18+25463317*h^34+ 28*h^48+10677*h^44+163429497*h^16+798*h^46+61539876*h^32+8717101*h^36)*t^12+h^ 49*(24+32716211*h^14+82904016*h^28+1132*h^2+716186*h^38+175443*h^6+149849*h^40+ 140814945*h^20+46749799*h^30+1034673*h^8+159985028*h^22+152966541*h^24+13422287 *h^12+122956370*h^26+4305222*h^10+24307*h^42+19124*h^4+104140769*h^18+8552151*h ^34+8*h^48+2873*h^44+64304431*h^16+219*h^46+21962569*h^32+2741494*h^36)*t^13-h^ 56*(29936227*h^16+11918125*h^28+978200*h^8+538854*h^34+2548*h^2+41436201*h^18+ 1859493*h^32+17791600*h^14+125039*h^36+22899*h^38+5203914*h^30+26*h^44+92+350*h ^42+22453284*h^26+44725552*h^22+47361498*h^20+h^46+3258*h^40+30261*h^4+34880871 *h^24+210115*h^6+8603583*h^12+3311356*h^10)*t^14+h^63*(59789*h^32+4086568*h^24+ 9825094*h^18+1509473*h^10+9079488*h^20+8574107*h^16+128+6*h^40+522650*h^8+11334 *h^34+1566*h^36+2628*h^2+242749*h^30+134917*h^6+6776589*h^22+1988300*h^26+ 6019737*h^14+24154*h^4+3384967*h^12+140*h^38+777536*h^28)*t^15-h^70*(868767*h^ 20+1210224*h^14+817150*h^12+15*h^34+1388753*h^16+475766*h^22+303*h^32+2826*h^30 +424151*h^10+76+9973*h^4+16385*h^28+1276*h^2+167110*h^8+66751*h^26+203133*h^24+ 1241909*h^18+48733*h^6)*t^16+h^77*(1+h^2)*(20*h^26+312*h^24+2145*h^22+8737*h^20 +23785*h^18+45784*h^16+63715*h^14+64373*h^12+46820*h^10+24030*h^8+8453*h^6+1944 *h^4+264*h^2+16)*t^17-h^86*(1+h^2)*(15*h^18+173*h^16+827*h^14+2194*h^12+3595*h^ 10+3741*h^8+2483*h^6+1032*h^4+243*h^2+24)*t^18+h^95*(1+h^2)*(6*h^10+42*h^8+105* h^6+125*h^4+67*h^2+12)*t^19-h^104*(2+h^2)*(1+h^2)*t^20)/(t*h^5-1)/(t^2*h^10+1-2 *h^3*t-2*t*h^5)/(t^2*h^10+1-h^3*t-2*t*h^5)/(1-2*t^3*h^13-8*t^3*h^15+t^4*h^20+4* t^2*h^8-2*h^3*t+h^6*t^2-8*t*h^5+6*t^2*h^10-9*t^3*h^17-2*t^3*h^19-9*t*h^7-2*t*h^ 9+20*t^2*h^14+8*t^2*h^16+16*t^2*h^12+t^2*h^18)/(t^4*h^20-4*t^3*h^15-6*t^3*h^13-\ 2*t^3*h^11+6*t^2*h^10+12*t^2*h^8+5*h^6*t^2-4*t*h^5-6*h^3*t-2*h*t+1)/(1-722*t^3* h^13-438*t^3*h^11-922*t^3*h^15+2074*t^4*h^20+152*t^2*h^8+1092*t^4*h^24+24*t^2*h ^4+4*t^2*h^2-20*h^3*t+62*h^6*t^2-4*t^7*h^31-34*t*h^5+320*t^2*h^10+33*t^4*h^12-4 *h*t+848*t^4*h^26-34*t^7*h^35-20*t^7*h^33-54*t^3*h^25+376*t^6*h^32-148*t^3*h^9-\ 20*t^3*h^7+t^8*h^40+221*t^6*h^34+62*t^6*h^26+24*t^6*h^24-1178*t^3*h^17+4*t^6*h^ 22+320*t^6*h^30-1224*t^3*h^19-796*t^3*h^21-22*t*h^7+152*t^6*h^28-4*t*h^9-722*t^ 5*h^23-922*t^5*h^25+221*t^2*h^14+60*t^2*h^16-20*t^5*h^17-290*t^3*h^23+376*t^2*h ^12+772*t^4*h^16-148*t^5*h^19+6*t^2*h^18-796*t^5*h^31-1224*t^5*h^29-1178*t^5*h^ 27-438*t^5*h^21-290*t^5*h^33-54*t^5*h^35-22*t^7*h^37-4*t^5*h^37+t^4*h^36+104*t^ 4*h^32+16*t^4*h^34+352*t^4*h^30+1720*t^4*h^22+232*t^4*h^14+676*t^4*h^28+1584*t^ 4*h^18+60*t^6*h^36+6*t^6*h^38-4*t^7*h^39-4*t^3*h^27) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 72 _Z + 338 _Z - 72 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 72 b + 338 b - 72 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0149322967114543 We have the following proven facts The total number of tilings of the region is asymptotic to: /4215091 21121393 4511863 2 62677 3\ n |------- - -------- b + ------- b - ----- b | (1/b) \ 46200 46200 46200 46200 / that in floating-point is: n 84.4308638498834 66.9689344729479 and in Maple input format: HFloat(84.4308638498833801)*HFloat(66.9689344729478790)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 14527 46573 b 52067 b 92 b /21031 1941 2665 2 31 3\ ----- + ------- - -------- + ----- + |----- + ---- b - ---- b + --- b | n 3080 2310 9240 1155 \4620 154 924 770 / and in Maple-input format: 14527/3080+46573/2310*b-52067/9240*b^2+92/1155*b^3+(21031/4620+1941/154*b-2665/ 924*b^2+31/770*b^3)*n and in floating point: 5.01635934124667 + 4.73972665245616 n The asymptotic expression for the variance, as a function of n is: 3 2 93529 b 4393 1219 b 406831 b /4789 292 521 2 3\ - ------- + ---- + ------- - --------- + |---- - --- b + --- b - 4/175 b | n 34650 1386 6930 34650 \1575 35 315 / and in Maple-input format: -93529/34650*b+4393/1386+1219/6930*b^3-406831/34650*b^2+(4789/1575-292/35*b+521 /315*b^2-4/175*b^3)*n and in floating point: 3.12662929303916 + 2.91642561768438 n The even alpha coefficients until the, 8, -th , are: 10071977 1860345 2663935 2 1359 3 - -------- - ------- b + -------- b - ------- b 31565996 7891499 31565996 1127357 / [1, 3 + ------------------------------------------------- + | n \ 4229721857689 31087513402069 282029554414063 2 2098744183369 3\ - ------------- + -------------- b - --------------- b + ------------- b | 267363986120 26736398612 53472797224 26736398612 / / 2 / n , 15 / 127691880753 13381073505 24298021815 2 12501351 3 - ------------ - ----------- b + ----------- b - --------- b 26736398612 6684099653 26736398612 954871379 / + -------------------------------------------------------------- + | n \ 10811342765266389 198721211446211427 3611459671931089383 2 - ----------------- + ------------------ b - ------------------- b 45291459248728 11322864812182 45291459248728 13437546113629557 3\ / 2 + ----------------- b | / n , 105 11322864812182 / / 127419293721 6252927570 14750749455 2 15473214 3 - ------------ - ---------- b + ----------- b - --------- b 1909742758 954871379 1909742758 136410197 / + ------------------------------------------------------------- + | n \ 10861484120411385 199919450902128189 3639700061518595631 2 - ----------------- + ------------------ b - ------------------- b 3235104232052 808776058013 3235104232052 13542702463821039 3\ / 2 + ----------------- b | / n ] 808776058013 / / and in Maple-input format: [1, 3+(-10071977/31565996-1860345/7891499*b+2663935/31565996*b^2-1359/1127357*b ^3)/n+(-4229721857689/267363986120+31087513402069/26736398612*b-282029554414063 /53472797224*b^2+2098744183369/26736398612*b^3)/n^2, 15+(-127691880753/ 26736398612-13381073505/6684099653*b+24298021815/26736398612*b^2-12501351/ 954871379*b^3)/n+(-10811342765266389/45291459248728+198721211446211427/ 11322864812182*b-3611459671931089383/45291459248728*b^2+13437546113629557/ 11322864812182*b^3)/n^2, 105+(-127419293721/1909742758-6252927570/954871379*b+ 14750749455/1909742758*b^2-15473214/136410197*b^3)/n+(-10861484120411385/ 3235104232052+199919450902128189/808776058013*b-3639700061518595631/ 3235104232052*b^2+13542702463821039/808776058013*b^3)/n^2] and in floating-point it is: 0.322578132263974 0.366549734211393 [1, 3 - ----------------- + -----------------, n 2 n 4.80564740984938 5.58676672715327 15 - ---------------- + ----------------, n 2 n 66.8167197023482 82.8924981567179 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 34073379 1025460972 238681809 2 2382372 3 ----------- + ---------- b - ---------- b + ---------- b 33420498265 6684099653 6684099653 4774356895 / 10244587213269 [---------------------------------------------------------- + |- -------------- n \ 56614324060910 346580913584418 3550140778981761 2 26428615884978 3\ / 2 + --------------- b - ---------------- b + -------------- b | / n , 28307162030455 56614324060910 28307162030455 / / 681467580 102546097200 23868180900 2 47647440 3 ---------- + ------------ b - ----------- b + --------- b 6684099653 6684099653 6684099653 954871379 / ----------------------------------------------------------- + | n \ 101221026247350 6821684616903285 5068274354397630 2 - --------------- + ---------------- b - ---------------- b 5661432406091 5661432406091 808776058013 528246105074085 3\ / 2 + --------------- b | / n , 5661432406091 / / 1533302055 230728718700 53703407025 2 107206740 3 ---------- + ------------ b - ----------- b + --------- b 136410197 136410197 136410197 19487171 / ----------------------------------------------------------- + | n \ 4954024363336695 2318243311667394765 12283384700649461265 2 - ---------------- + ------------------- b - -------------------- b 2541867610898 17793073276286 17793073276286 182902348381198365 3\ / 2 + ------------------ b | / n ] 17793073276286 / / and in Maple-input format: [(34073379/33420498265+1025460972/6684099653*b-238681809/6684099653*b^2+2382372 /4774356895*b^3)/n+(-10244587213269/56614324060910+346580913584418/ 28307162030455*b-3550140778981761/56614324060910*b^2+26428615884978/ 28307162030455*b^3)/n^2, (681467580/6684099653+102546097200/6684099653*b-\ 23868180900/6684099653*b^2+47647440/954871379*b^3)/n+(-101221026247350/ 5661432406091+6821684616903285/5661432406091*b-5068274354397630/808776058013*b^ 2+528246105074085/5661432406091*b^3)/n^2, (1533302055/136410197+230728718700/ 136410197*b-53703407025/136410197*b^2+107206740/19487171*b^3)/n+(-\ 4954024363336695/2541867610898+2318243311667394765/17793073276286*b-\ 12283384700649461265/17793073276286*b^2+182902348381198365/17793073276286*b^3)/ n^2] and in floating-point it is: 0.00330245744902289 0.0121082504372843 0.330245744902289 1.28350631922617 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 36.4095933740273 157.349640550966 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 37, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 2, d[1] = 3, d[2] = 3 Then infinity ----- \ n 2 6 8 12 10 4 2 ) A(n, h) t = (-1 - 18 h - 108 h - 67 h - h - 15 h - 71 h + h / ----- n = 0 6 2 16 4 8 18 20 (2596 h + 154 h + 8 + 517 h + 894 h + 4511 h + 75 h + 4 h 14 12 10 4 8 2 4 + 1832 h + 3823 h + 5097 h ) t - h (34258 h + 336 h + 2620 h + 16 28 16 6 24 14 12 + 6 h + 81160 h + 11676 h + 1160 h + 105586 h + 100981 h 20 22 10 18 26 2 8 + 19638 h + 5894 h + 69778 h + 46442 h + 130 h ) t + h ( 12 4 34 2 30 32 6 734214 h + 17880 h + 4 h + 2560 h + 1020 h + 96 h + 74016 h 22 28 20 14 26 + 268900 h + 6618 h + 160 + 543500 h + 974097 h + 30114 h 10 18 24 8 16 3 12 + 442342 h + 848232 h + 102676 h + 209410 h + 1026730 h ) t - h 2 38 36 34 32 30 28 (2 + h ) (h + 25 h + 299 h + 2340 h + 13241 h + 55786 h 26 24 22 20 18 + 177899 h + 438396 h + 853356 h + 1334498 h + 1695642 h 16 14 12 10 8 + 1766314 h + 1517971 h + 1073524 h + 614387 h + 276058 h 6 4 2 4 16 22 + 93260 h + 22168 h + 3280 h + 224) t + h (12517011 h 30 40 4 12 32 + 1225730 h + 446 h + 48352 h + 384 + 5544028 h + 416188 h 2 20 24 42 6 44 + 6016 h + 14587485 h + 9232566 h + 31 h + 253568 h + h 18 28 26 36 8 + 14759480 h + 2932697 h + 5728606 h + 24473 h + 942824 h 10 16 38 14 34 5 + 2605896 h + 12830194 h + 3955 h + 9345556 h + 113795 h ) t - 22 42 44 20 12 18 h (118 h + 4 h + 26042936 h + 10443784 h + 26669891 h 26 36 4 40 28 + 10024070 h + 76004 h + 136576 h + 1604 h + 5538360 h 30 24 8 16 38 + 2612771 h + 15807079 h + 2070368 h + 23219940 h + 13328 h 14 2 22 34 6 + 17059324 h + 19072 h + 21794596 h + 317720 h + 627104 h 10 32 6 28 4 26 + 5219752 h + 1019814 h + 1280) t + h (156256 h + 10771433 h 42 14 36 16 24 + 170 h + 18208464 h + 94954 h + 24912976 h + 16914512 h 12 10 2 6 44 32 + 11123036 h + 5613704 h + 21632 h + 707840 h + 6 h + 1166130 h 28 40 38 34 8 + 6011396 h + 2208 h + 17484 h + 378718 h + 2277864 h 30 20 22 18 7 + 2893588 h + 1408 + 27904954 h + 23303714 h + 28648954 h ) t - 34 8 42 40 24 26 44 h (1240688 h + 105 h + 1266 h + 11479867 h + 7254549 h + 4 h 32 34 30 38 6 + 512 + 640088 h + 198063 h + 1716751 h + 9425 h + 346496 h 28 22 10 4 18 + 3855282 h + 15435943 h + 3331760 h + 68736 h + 17985041 h 36 2 14 16 12 + 49422 h + 8640 h + 11486471 h + 15628614 h + 6930952 h 20 8 42 28 24 14 + 17877816 h ) t + h (563590 h + 2728120 h + 5819596 h 22 4 38 16 6 + 4498012 h + 79040 h + 254 h + 7086087 h + 768 + 332792 h 36 34 10 40 42 12 + 1879 h + 10747 h + 2215818 h + 23 h + h + 3969682 h 30 26 32 20 8 + 186920 h + 1371535 h + 49823 h + 6213527 h + 985168 h 18 2 9 50 8 30 32 + 7230477 h + 11552 h ) t - h (371084 h + 6222 h + 1028 h 12 20 24 34 26 + 1221546 h + 1021482 h + 253778 h + 106 h + 92305 h 10 4 18 14 16 + 756306 h + 384 + 35056 h + 1476760 h + 1602887 h + 1712266 h 22 6 28 2 36 10 58 2 + 566760 h + 136956 h + 27130 h + 5456 h + 5 h ) t + h (1 + h ) 28 26 24 22 20 18 16 (10 h + 174 h + 1334 h + 6104 h + 19096 h + 44095 h + 78173 h 14 12 10 8 6 4 + 107593 h + 113563 h + 89498 h + 51056 h + 20362 h + 5400 h 2 11 68 18 16 14 12 10 + 864 h + 64) t - h (10 h + 126 h + 654 h + 1888 h + 3410 h 8 6 4 2 2 2 12 + 4028 h + 3180 h + 1565 h + 424 h + 48) (1 + h ) t 78 10 8 6 4 2 2 2 13 + h (5 h + 39 h + 107 h + 138 h + 70 h + 12) (1 + h ) t 88 2 2 4 14 / 6 - h (1 + h ) (1 + 3 h + h ) t ) / ((t h - 1) / 2 12 2 4 6 2 12 4 6 (t h + 1 - 4 h t - 8 t h - 2 t h ) (t h + 1 - 2 t h - 2 t h ) 2 12 6 8 10 8 4 2 8 (t h + 1 - 2 t h - 4 t h - t h ) (1 - 10 t h - 4 t h + 4 t h 4 24 2 20 2 10 2 14 2 16 3 20 + t h + t h + 8 t h + 16 t h + 20 t h - 10 t h 3 18 3 16 10 6 2 12 2 18 - 12 t h - 4 t h - 2 t h - 12 t h + 6 t h + 8 t h 3 22 8 4 2 8 4 24 2 2 4 - 2 t h ) (1 - 16 t h - 32 t h + 80 t h + t h - 8 h t + 16 t h 6 2 2 20 2 10 2 14 2 16 3 20 + 64 h t + t h + 32 t h + 16 t h + 20 t h - 16 t h 3 18 3 16 10 6 2 12 3 14 - 40 t h - 32 t h - 2 t h - 40 t h + 6 t h - 8 t h 2 18 3 22 + 8 t h - 2 t h )) and in Maple input format: (-1-18*h^2-108*h^6-67*h^8-h^12-15*h^10-71*h^4+h^2*(2596*h^6+154*h^2+8+517*h^16+ 894*h^4+4511*h^8+75*h^18+4*h^20+1832*h^14+3823*h^12+5097*h^10)*t-h^4*(34258*h^8 +336*h^2+2620*h^4+16+6*h^28+81160*h^16+11676*h^6+1160*h^24+105586*h^14+100981*h ^12+19638*h^20+5894*h^22+69778*h^10+46442*h^18+130*h^26)*t^2+h^8*(734214*h^12+ 17880*h^4+4*h^34+2560*h^2+1020*h^30+96*h^32+74016*h^6+268900*h^22+6618*h^28+160 +543500*h^20+974097*h^14+30114*h^26+442342*h^10+848232*h^18+102676*h^24+209410* h^8+1026730*h^16)*t^3-h^12*(2+h^2)*(h^38+25*h^36+299*h^34+2340*h^32+13241*h^30+ 55786*h^28+177899*h^26+438396*h^24+853356*h^22+1334498*h^20+1695642*h^18+ 1766314*h^16+1517971*h^14+1073524*h^12+614387*h^10+276058*h^8+93260*h^6+22168*h ^4+3280*h^2+224)*t^4+h^16*(12517011*h^22+1225730*h^30+446*h^40+48352*h^4+384+ 5544028*h^12+416188*h^32+6016*h^2+14587485*h^20+9232566*h^24+31*h^42+253568*h^6 +h^44+14759480*h^18+2932697*h^28+5728606*h^26+24473*h^36+942824*h^8+2605896*h^ 10+12830194*h^16+3955*h^38+9345556*h^14+113795*h^34)*t^5-h^22*(118*h^42+4*h^44+ 26042936*h^20+10443784*h^12+26669891*h^18+10024070*h^26+76004*h^36+136576*h^4+ 1604*h^40+5538360*h^28+2612771*h^30+15807079*h^24+2070368*h^8+23219940*h^16+ 13328*h^38+17059324*h^14+19072*h^2+21794596*h^22+317720*h^34+627104*h^6+5219752 *h^10+1019814*h^32+1280)*t^6+h^28*(156256*h^4+10771433*h^26+170*h^42+18208464*h ^14+94954*h^36+24912976*h^16+16914512*h^24+11123036*h^12+5613704*h^10+21632*h^2 +707840*h^6+6*h^44+1166130*h^32+6011396*h^28+2208*h^40+17484*h^38+378718*h^34+ 2277864*h^8+2893588*h^30+1408+27904954*h^20+23303714*h^22+28648954*h^18)*t^7-h^ 34*(1240688*h^8+105*h^42+1266*h^40+11479867*h^24+7254549*h^26+4*h^44+512+640088 *h^32+198063*h^34+1716751*h^30+9425*h^38+346496*h^6+3855282*h^28+15435943*h^22+ 3331760*h^10+68736*h^4+17985041*h^18+49422*h^36+8640*h^2+11486471*h^14+15628614 *h^16+6930952*h^12+17877816*h^20)*t^8+h^42*(563590*h^28+2728120*h^24+5819596*h^ 14+4498012*h^22+79040*h^4+254*h^38+7086087*h^16+768+332792*h^6+1879*h^36+10747* h^34+2215818*h^10+23*h^40+h^42+3969682*h^12+186920*h^30+1371535*h^26+49823*h^32 +6213527*h^20+985168*h^8+7230477*h^18+11552*h^2)*t^9-h^50*(371084*h^8+6222*h^30 +1028*h^32+1221546*h^12+1021482*h^20+253778*h^24+106*h^34+92305*h^26+756306*h^ 10+384+35056*h^4+1476760*h^18+1602887*h^14+1712266*h^16+566760*h^22+136956*h^6+ 27130*h^28+5456*h^2+5*h^36)*t^10+h^58*(1+h^2)*(10*h^28+174*h^26+1334*h^24+6104* h^22+19096*h^20+44095*h^18+78173*h^16+107593*h^14+113563*h^12+89498*h^10+51056* h^8+20362*h^6+5400*h^4+864*h^2+64)*t^11-h^68*(10*h^18+126*h^16+654*h^14+1888*h^ 12+3410*h^10+4028*h^8+3180*h^6+1565*h^4+424*h^2+48)*(1+h^2)^2*t^12+h^78*(5*h^10 +39*h^8+107*h^6+138*h^4+70*h^2+12)*(1+h^2)^2*t^13-h^88*(1+h^2)*(1+3*h^2+h^4)*t^ 14)/(t*h^6-1)/(t^2*h^12+1-4*h^2*t-8*t*h^4-2*t*h^6)/(t^2*h^12+1-2*t*h^4-2*t*h^6) /(t^2*h^12+1-2*t*h^6-4*t*h^8-t*h^10)/(1-10*t*h^8-4*t*h^4+4*t^2*h^8+t^4*h^24+t^2 *h^20+8*t^2*h^10+16*t^2*h^14+20*t^2*h^16-10*t^3*h^20-12*t^3*h^18-4*t^3*h^16-2*t *h^10-12*t*h^6+6*t^2*h^12+8*t^2*h^18-2*t^3*h^22)/(1-16*t*h^8-32*t*h^4+80*t^2*h^ 8+t^4*h^24-8*h^2*t+16*t^2*h^4+64*h^6*t^2+t^2*h^20+32*t^2*h^10+16*t^2*h^14+20*t^ 2*h^16-16*t^3*h^20-40*t^3*h^18-32*t^3*h^16-2*t*h^10-40*t*h^6+6*t^2*h^12-8*t^3*h ^14+8*t^2*h^18-2*t^3*h^22) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 98 _Z + 243 _Z + _Z - 98 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 98 b + 243 b + b - 98 b and whose floating-point appx. , to, 10, digits is, 0.0104750075324274 We have the following proven facts The total number of tilings of the region is asymptotic to: /3682219 691351 503947 2 77143 3\ n |------- - ------ b + ------ b - ----- b | (1/b) \ 15120 1120 2016 30240 / that in floating-point is: n 237.094439896317 95.4653251469566 and in Maple input format: HFloat(237.094439896316914)*HFloat(95.4653251469566015)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 4709 43594 b 22366 b 51 b /587 136 2 3\ ---- - ------- + -------- - ----- + |--- - 32/3 b + --- b - 1/15 b | n 630 315 315 70 \105 21 / and in Maple-input format: 4709/630-43594/315*b+22366/315*b^2-51/70*b^3+(587/105-32/3*b+136/21*b^2-1/15*b^ 3)*n and in floating point: 6.03271868200705 + 5.47945330476514 n The asymptotic expression for the variance, as a function of n is: 2 3 3423452 b 89687 3901 b 6851708 b ---------- + ----- - ------- - --------- 4725 4725 525 4725 /14402 2336 928 2 142 3\ + |----- - ---- b + --- b - ---- b | n \4725 315 315 4725 / and in Maple-input format: 3423452/4725*b^2+89687/4725-3901/525*b^3-6851708/4725*b+(14402/4725-2336/315*b+ 928/315*b^2-142/4725*b^3)*n and in floating point: 3.87109157260827 + 2.97068422339654 n The even alpha coefficients until the, 8, -th , are: 96251 65090 266590 2 9403 3 ----- - ----- b + ------ b - ---- b 48552 289 2023 6936 [1, 3 + ------------------------------------- n 8416290441443 40384668489073 20192495665177 2 276114384671 3 ------------- - -------------- b + -------------- b - ------------ b 8253840 412692 412692 550256 + ----------------------------------------------------------------------, 2 n 13328593 142489470 83178510 2 6845303 3 - -------- + --------- b - -------- b + ------- b 275128 34391 34391 275128 /104352277997537 15 + --------------------------------------------------- + |--------------- n \ 9354352 625903781925228 312940403894232 2 51350124829279 3\ / 2 - --------------- b + --------------- b - -------------- b | / n , 105 584647 584647 9354352 / / 34838521 802329180 468674940 2 19285391 3 - -------- + --------- b - --------- b + -------- b 19652 4913 4913 19652 / + ----------------------------------------------------- + | n \ 65643031883307 787451400206602 393655338908338 2 32297186800223 3 -------------- - --------------- b + --------------- b - -------------- b 668168 83521 83521 668168 \ / 2 | / n ] / / and in Maple-input format: [1, 3+(96251/48552-65090/289*b+266590/2023*b^2-9403/6936*b^3)/n+(8416290441443/ 8253840-40384668489073/412692*b+20192495665177/412692*b^2-276114384671/550256*b ^3)/n^2, 15+(-13328593/275128+142489470/34391*b-83178510/34391*b^2+6845303/ 275128*b^3)/n+(104352277997537/9354352-625903781925228/584647*b+312940403894232 /584647*b^2-51350124829279/9354352*b^3)/n^2, 105+(-34838521/19652+802329180/ 4913*b-468674940/4913*b^2+19285391/19652*b^3)/n+(65643031883307/668168-\ 787451400206602/83521*b+393655338908338/83521*b^2-32297186800223/668168*b^3)/n^ 2] and in floating-point it is: 0.362343402865160 0.520938290650992 [1, 3 - ----------------- + -----------------, n 2 n 5.31015954566227 7.67888376659130 15 - ---------------- + ----------------, n 2 n 72.5923529568597 109.391087140240 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 2688741 25867512 15115896 2 1555011 3 - ------- + -------- b - -------- b + ------- b 343910 34391 34391 343910 / [------------------------------------------------- + | n \ 4840306335199 116128582587176 58072379964944 2 2382270835057 3\ - ------------- + --------------- b - -------------- b + ------------- b | 11692940 2923235 2923235 11692940 / 26887410 2586751200 1511589600 2 15550110 3 - -------- + ---------- b - ---------- b + -------- b / 2 34391 34391 34391 34391 / / n , ------------------------------------------------------- + | / n \ 48391072480825 2321996842414195 1161214825272955 2 - -------------- + ---------------- b - ---------------- b 1169294 584647 584647 23817957860245 3\ / 2 + -------------- b | / n , 1169294 / / 846953415 40741331400 23807536200 2 489828465 3 - --------- + ----------- b - ----------- b + --------- b 9826 4913 4913 9826 / ----------------------------------------------------------- + | n \ 12942293473295865 1242047184105192885 621270835418189565 2 - ----------------- + ------------------- b - ------------------ b 2839714 2839714 2839714 3185773478359290 3\ / 2 + ---------------- b | / n ] 1419857 / / and in Maple-input format: [(-2688741/343910+25867512/34391*b-15115896/34391*b^2+1555011/343910*b^3)/n+(-\ 4840306335199/11692940+116128582587176/2923235*b-58072379964944/2923235*b^2+ 2382270835057/11692940*b^3)/n^2, (-26887410/34391+2586751200/34391*b-1511589600 /34391*b^2+15550110/34391*b^3)/n+(-48391072480825/1169294+2321996842414195/ 584647*b-1161214825272955/584647*b^2+23817957860245/1169294*b^3)/n^2, (-\ 846953415/9826+40741331400/4913*b-23807536200/4913*b^2+489828465/9826*b^3)/n+(-\ 12942293473295865/2839714+1242047184105192885/2839714*b-621270835418189565/ 2839714*b^2+3185773478359290/1419857*b^3)/n^2] and in floating-point it is: 0.0124991497315142 0.0544922377618646 1.24991497315141 5.71505363902001 [------------------ - ------------------, ---------------- - ----------------, n 2 n 2 n n 137.803125914933 693.363965786245 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 38, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 3, d[1] = 1, d[2] = 1 Then infinity ----- \ n 2 2 ) A(n, h) t = ((1 + 3 h ) (1 + h ) / ----- n = 0 2 6 2 8 4 - h (12 h + 26 h + 6 + 2 h + 29 h ) t 4 4 2 6 8 2 + h (85 h + 15 + 65 h + 47 h + 9 h ) t 6 8 6 4 2 3 - h (20 + 12 h + 58 h + 105 h + 80 h ) t 8 2 6 4 2 4 10 2 2 2 5 + h (1 + h ) (4 h + 21 h + 35 h + 15) t - 2 h (h + 3) (1 + h ) t 12 2 6 / 2 2 4 2 4 3 10 + h (1 + h ) t ) / ((h t - 1) (t h + 1 - 2 h t - 2 t h ) (2 t h / 4 8 3 8 2 8 3 6 6 2 6 2 4 - t h + 6 t h - 5 t h + 4 t h - 12 h t + 2 t h - 6 t h 4 2 + 6 t h + 4 h t - 1)) and in Maple input format: ((1+3*h^2)*(1+h^2)-h^2*(12*h^6+26*h^2+6+2*h^8+29*h^4)*t+h^4*(85*h^4+15+65*h^2+ 47*h^6+9*h^8)*t^2-h^6*(20+12*h^8+58*h^6+105*h^4+80*h^2)*t^3+h^8*(1+h^2)*(4*h^6+ 21*h^4+35*h^2+15)*t^4-2*h^10*(h^2+3)*(1+h^2)^2*t^5+h^12*(1+h^2)*t^6)/(h^2*t-1)/ (t^2*h^4+1-2*h^2*t-2*t*h^4)/(2*t^3*h^10-t^4*h^8+6*t^3*h^8-5*t^2*h^8+4*t^3*h^6-\ 12*h^6*t^2+2*t*h^6-6*t^2*h^4+6*t*h^4+4*h^2*t-1) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 12 _Z + 23 _Z - 12 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 12 b + 23 b - 12 b + 1 and whose floating-point appx. , to, 10, digits is, 0.102347484250603 We have the following proven facts The total number of tilings of the region is asymptotic to: /1471 310 166 2 139 3\ n |---- - --- b + --- b - --- b | (1/b) \210 21 21 210 / that in floating-point is: n 5.57601090844784 9.77063586195681 and in Maple input format: HFloat(5.57601090844783887)*HFloat(9.77063586195681388)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 337 48 b 22 b 9 b /472 44 2 3\ --- - ---- + ----- - ---- + |--- - 32/7 b + -- b - 6/35 b | n 105 7 7 35 \105 21 / and in Maple-input format: 337/105-48/7*b+22/7*b^2-9/35*b^3+(472/105-32/7*b+44/21*b^2-6/35*b^3)*n and in floating point: 2.54035826121161 + 4.04912772914107 n The asymptotic expression for the variance, as a function of n is: 3 2 3212 208 b 46 b 328 b / 416 656 2 92 3 2944\ ---- - ----- - ----- + ------ + |- --- b + --- b - --- b + ----| n 1575 35 175 105 \ 105 315 525 1575/ and in Maple-input format: 3212/1575-208/35*b-46/175*b^3+328/105*b^2+(-416/105*b+656/315*b^2-92/525*b^3+ 2944/1575)*n and in floating point: 1.46356872276963 + 1.48534211151308 n The even alpha coefficients until the, 8, -th , are: 138536 1500360 1306220 2 118119 3 ------ - ------- b + ------- b - ------ b 2023 2023 2023 2023 [1, 3 + ------------------------------------------- n 178709207 368140044 188363694 2 30921315 3 --------- - --------- b + --------- b - -------- b 171955 34391 34391 68782 + ----------------------------------------------------, 15 2 n 44045544 468024840 406709580 2 36770751 3 - -------- + --------- b - --------- b + -------- b 34391 34391 34391 34391 + ----------------------------------------------------- n 33168240255 342705312240 183068876070 2 30364686759 3 - ----------- + ------------ b - ------------ b + ----------- b 584647 584647 584647 1169294 + -----------------------------------------------------------------, 105 2 n 246835536 2637282960 2293010520 2 207323694 3 - --------- + ---------- b - ---------- b + --------- b 4913 4913 4913 4913 + --------------------------------------------------------- n 131028594897 1353036161760 716230164780 2 59271740223 3 - ------------ + ------------- b - ------------ b + ----------- b 83521 83521 83521 83521 + -------------------------------------------------------------------] 2 n and in Maple-input format: [1, 3+(138536/2023-1500360/2023*b+1306220/2023*b^2-118119/2023*b^3)/n+( 178709207/171955-368140044/34391*b+188363694/34391*b^2-30921315/68782*b^3)/n^2, 15+(-44045544/34391+468024840/34391*b-406709580/34391*b^2+36770751/34391*b^3)/n +(-33168240255/584647+342705312240/584647*b-183068876070/584647*b^2+30364686759 /1169294*b^3)/n^2, 105+(-246835536/4913+2637282960/4913*b-2293010520/4913*b^2+ 207323694/4913*b^3)/n+(-131028594897/83521+1353036161760/83521*b-716230164780/ 83521*b^2+59271740223/83521*b^3)/n^2] and in floating-point it is: 0.724686808934144 0.586488850215835 [1, 3 - ----------------- + -----------------, n 2 n 10.6203187765834 9.26248996964857 15 - ---------------- + ----------------, n 2 n 145.184702468171 147.037129145152 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 39686112 85061664 73979568 2 33445548 3 - -------- + -------- b - -------- b + -------- b 171955 34391 34391 171955 [--------------------------------------------------- n 18344930292 37872665304 19954468872 2 8247533466 3 - ----------- + ----------- b - ----------- b + ---------- b 2923235 584647 584647 2923235 + --------------------------------------------------------------, 2 n 793722240 8506166400 7397956800 2 668910960 3 - --------- + ---------- b - ---------- b + --------- b 34391 34391 34391 34391 --------------------------------------------------------- n 269504728560 2779893178800 1445970513600 2 119160971520 3 - ------------ + ------------- b - ------------- b + ------------ b 584647 584647 584647 584647 + ---------------------------------------------------------------------, 2 n 12501125280 133972120800 116517819600 2 10535347620 3 - ----------- + ------------ b - ------------ b + ----------- b 4913 4913 4913 4913 / ----------------------------------------------------------------- + | n \ 180537137515980 1869366969187200 1038486868335000 2 --------------- - ---------------- b + ---------------- b 1419857 1419857 1419857 86898439321470 3\ / 2 - -------------- b | / n ] 1419857 / / and in Maple-input format: [(-39686112/171955+85061664/34391*b-73979568/34391*b^2+33445548/171955*b^3)/n+( -18344930292/2923235+37872665304/584647*b-19954468872/584647*b^2+8247533466/ 2923235*b^3)/n^2, (-793722240/34391+8506166400/34391*b-7397956800/34391*b^2+ 668910960/34391*b^3)/n+(-269504728560/584647+2779893178800/584647*b-\ 1445970513600/584647*b^2+119160971520/584647*b^3)/n^2, (-12501125280/4913+ 133972120800/4913*b-116517819600/4913*b^2+10535347620/4913*b^3)/n+( 180537137515980/1419857-1869366969187200/1419857*b+1038486868335000/1419857*b^2 -86898439321470/1419857*b^3)/n^2] and in floating-point it is: 0.0249983457428914 0.117735852541510 2.49983457428517 14.4632647162786 [------------------ - -----------------, ---------------- - ----------------, n 2 n 2 n n 275.605844315037 2024.30307778587 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 39, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 3, d[1] = 1, d[2] = 3 Then infinity ----- \ n 2 4 4 2 2 ) A(n, h) t = (-(1 + 3 h + h ) (9 h + 9 h + 1) + h / ----- n = 0 12 8 6 10 2 14 4 (290 h + 1482 h + 1369 h + 901 h + 160 h + 38 h + 12 + 683 h ) t 4 2 18 16 14 12 10 - h (1 + h ) (52 h + 560 h + 2669 h + 7300 h + 12448 h 8 6 4 2 2 6 2 + 13569 h + 9348 h + 3838 h + 812 h + 60) t + h (2520 h 14 4 8 10 22 26 + 214903 h + 16028 h + 136379 h + 222031 h + 3376 h + 24 h 6 20 12 18 24 16 + 57982 h + 16722 h + 257445 h + 55553 h + 416 h + 129240 h 3 8 30 14 18 12 + 160) t - h (32 h + 1582719 h + 934994 h + 1358216 h 16 26 2 22 8 28 + 1397020 h + 7476 h + 4160 h + 172514 h + 413108 h + 732 h 24 4 6 10 20 4 + 44413 h + 31224 h + 138832 h + 240 + 873735 h + 469510 h ) t + 10 2 30 28 26 24 22 h (1 + h ) (384 h + 6430 h + 49794 h + 232583 h + 727295 h 20 18 16 14 12 + 1615142 h + 2650771 h + 3302566 h + 3167760 h + 2340365 h 10 8 6 4 2 5 12 + 1314526 h + 546780 h + 161632 h + 31824 h + 3712 h + 192) t - h 2 32 30 28 26 24 (1 + h ) (1952 h + 29369 h + 202930 h + 853612 h + 2445678 h 22 20 18 16 14 + 5068414 h + 7889083 h + 9460802 h + 8886233 h + 6582857 h 12 10 8 6 4 2 + 3828954 h + 1718192 h + 575924 h + 136992 h + 21280 h + 1856 h 6 16 8 2 22 30 + 64) t + h (384 + 1018536 h + 6976 h + 3876692 h + 5496 h 4 10 6 16 12 + 57888 h + 2612628 h + 292784 h + 9756369 h + 5153346 h 20 28 14 18 24 + 6932564 h + 72581 h + 7982091 h + 9353738 h + 1573249 h 26 2 2 7 20 2 32 30 + 434542 h ) (1 + h ) t - h (1 + h ) (9414 h + 129667 h 28 26 24 22 20 + 816518 h + 3137134 h + 8284514 h + 16054287 h + 23769980 h 18 16 14 12 10 + 27586166 h + 25486087 h + 18889635 h + 11243718 h + 5343380 h 8 6 4 2 8 24 2 + 1996288 h + 567792 h + 115488 h + 14848 h + 896) t + h (1 + h ) ( 32 30 28 26 24 10154 h + 138677 h + 866686 h + 3308791 h + 8693377 h 22 20 18 16 14 + 16782126 h + 24784507 h + 28725766 h + 26529668 h + 19670011 h 12 10 8 6 4 + 11721814 h + 5584692 h + 2096240 h + 601008 h + 123936 h 2 9 28 2 32 30 28 + 16320 h + 1024) t - h (1 + h ) (6900 h + 95724 h + 608090 h 26 24 22 20 18 + 2361200 h + 6309404 h + 12370246 h + 18497082 h + 21611586 h 16 14 12 10 8 + 20029533 h + 14847323 h + 8814306 h + 4156940 h + 1524820 h 6 4 2 10 32 2 32 + 418768 h + 80928 h + 9856 h + 576) t + h (1 + h ) (2856 h 30 28 26 24 22 + 41568 h + 277171 h + 1128145 h + 3146061 h + 6390401 h 20 18 16 14 12 + 9812498 h + 11671801 h + 10926750 h + 8110263 h + 4758518 h 10 8 6 4 2 11 + 2174140 h + 752120 h + 188432 h + 31776 h + 3136 h + 128) t - 38 10 18 14 20 22 h (2590180 h + 8125516 h + 7446592 h + 5917899 h + 3327213 h 26 24 28 30 16 + 448 + 429953 h + 1406554 h + 89384 h + 11296 h + 8753727 h 32 6 12 2 8 4 12 + 656 h + 295896 h + 4983921 h + 7472 h + 1022056 h + 59472 h ) t 44 30 12 16 26 6 + h (64 h + 2048303 h + 2132859 h + 14224 h + 234308 h 18 4 28 20 10 + 1464932 h + 672 + 58160 h + 1456 h + 763966 h + 1339069 h 22 2 24 8 14 13 50 + 294390 h + 9056 h + 79990 h + 655838 h + 2385202 h ) t - h 2 24 22 20 18 16 14 (1 + h ) (64 h + 992 h + 6860 h + 28494 h + 79599 h + 156619 h 12 10 8 6 4 2 + 220448 h + 222607 h + 160623 h + 81510 h + 27896 h + 5820 h 14 56 12 22 20 16 6 + 560) t + h (41997 h + 16 h + 264 h + 8352 h + 28581 h + 280 4 18 10 2 8 14 15 + 11358 h + 1952 h + 53361 h + 2668 h + 47163 h + 22835 h ) t - 62 6 14 4 10 2 12 h (84 + 3633 h + 244 h + 2043 h + 2621 h + 637 h + 1068 h 16 8 16 68 2 2 2 3 17 + 24 h + 3917 h ) t + 2 h (3 h + 7) (2 h + 1) (1 + h ) t 74 2 2 18 / 4 - h (2 h + 1) (1 + h ) t ) / ((t h - 1) / 2 8 2 4 8 4 2 8 2 10 (t h + 1 - 2 h t - 2 t h ) (1 - 2 t h - 4 t h + 6 t h + 12 t h 3 16 6 2 12 3 14 3 12 4 16 4 - 2 t h - 6 t h + 5 t h - 6 t h - 4 t h + t h ) (1 - 8 t h 2 8 2 2 4 6 2 2 10 6 2 12 + 6 t h - 4 h t + 4 t h + 8 h t + 8 t h - 4 t h + 4 t h 3 14 3 12 3 10 4 16 4 20 8 - 4 t h - 8 t h - 4 t h + t h ) (1 + 516 t h - 4 t h 4 2 8 4 24 2 2 4 6 2 - 20 t h + 144 t h + 33 t h - 8 h t + 24 t h + 96 h t 7 28 4 8 2 10 4 12 3 6 5 22 - 20 t h + 16 t h + 112 t h + 144 t h - 32 t h - 424 t h 6 32 6 26 7 32 6 24 5 16 6 22 + 4 t h + 112 t h - 4 t h + 144 t h - 144 t h + 96 t h 6 30 5 26 6 28 5 14 2 14 2 16 + 24 t h - 128 t h + 62 t h - 32 t h + 24 t h + 4 t h 3 20 3 18 3 16 4 10 6 2 12 - 20 t h - 128 t h - 320 t h + 64 t h - 16 t h + 62 t h 3 14 3 12 3 10 4 16 3 8 7 26 - 424 t h - 380 t h - 280 t h + 662 t h - 144 t h - 8 t h 5 18 5 28 4 22 8 32 7 30 5 20 - 280 t h - 20 t h + 200 t h + t h - 16 t h - 380 t h 6 20 4 14 4 18 5 24 + 24 t h + 352 t h + 752 t h - 320 t h )) and in Maple input format: (-(1+3*h^2+h^4)*(9*h^4+9*h^2+1)+h^2*(290*h^12+1482*h^8+1369*h^6+901*h^10+160*h^ 2+38*h^14+12+683*h^4)*t-h^4*(1+h^2)*(52*h^18+560*h^16+2669*h^14+7300*h^12+12448 *h^10+13569*h^8+9348*h^6+3838*h^4+812*h^2+60)*t^2+h^6*(2520*h^2+214903*h^14+ 16028*h^4+136379*h^8+222031*h^10+3376*h^22+24*h^26+57982*h^6+16722*h^20+257445* h^12+55553*h^18+416*h^24+129240*h^16+160)*t^3-h^8*(32*h^30+1582719*h^14+934994* h^18+1358216*h^12+1397020*h^16+7476*h^26+4160*h^2+172514*h^22+413108*h^8+732*h^ 28+44413*h^24+31224*h^4+138832*h^6+240+873735*h^10+469510*h^20)*t^4+h^10*(1+h^2 )*(384*h^30+6430*h^28+49794*h^26+232583*h^24+727295*h^22+1615142*h^20+2650771*h ^18+3302566*h^16+3167760*h^14+2340365*h^12+1314526*h^10+546780*h^8+161632*h^6+ 31824*h^4+3712*h^2+192)*t^5-h^12*(1+h^2)*(1952*h^32+29369*h^30+202930*h^28+ 853612*h^26+2445678*h^24+5068414*h^22+7889083*h^20+9460802*h^18+8886233*h^16+ 6582857*h^14+3828954*h^12+1718192*h^10+575924*h^8+136992*h^6+21280*h^4+1856*h^2 +64)*t^6+h^16*(384+1018536*h^8+6976*h^2+3876692*h^22+5496*h^30+57888*h^4+ 2612628*h^10+292784*h^6+9756369*h^16+5153346*h^12+6932564*h^20+72581*h^28+ 7982091*h^14+9353738*h^18+1573249*h^24+434542*h^26)*(1+h^2)^2*t^7-h^20*(1+h^2)* (9414*h^32+129667*h^30+816518*h^28+3137134*h^26+8284514*h^24+16054287*h^22+ 23769980*h^20+27586166*h^18+25486087*h^16+18889635*h^14+11243718*h^12+5343380*h ^10+1996288*h^8+567792*h^6+115488*h^4+14848*h^2+896)*t^8+h^24*(1+h^2)*(10154*h^ 32+138677*h^30+866686*h^28+3308791*h^26+8693377*h^24+16782126*h^22+24784507*h^ 20+28725766*h^18+26529668*h^16+19670011*h^14+11721814*h^12+5584692*h^10+2096240 *h^8+601008*h^6+123936*h^4+16320*h^2+1024)*t^9-h^28*(1+h^2)*(6900*h^32+95724*h^ 30+608090*h^28+2361200*h^26+6309404*h^24+12370246*h^22+18497082*h^20+21611586*h ^18+20029533*h^16+14847323*h^14+8814306*h^12+4156940*h^10+1524820*h^8+418768*h^ 6+80928*h^4+9856*h^2+576)*t^10+h^32*(1+h^2)*(2856*h^32+41568*h^30+277171*h^28+ 1128145*h^26+3146061*h^24+6390401*h^22+9812498*h^20+11671801*h^18+10926750*h^16 +8110263*h^14+4758518*h^12+2174140*h^10+752120*h^8+188432*h^6+31776*h^4+3136*h^ 2+128)*t^11-h^38*(2590180*h^10+8125516*h^18+7446592*h^14+5917899*h^20+3327213*h ^22+448+429953*h^26+1406554*h^24+89384*h^28+11296*h^30+8753727*h^16+656*h^32+ 295896*h^6+4983921*h^12+7472*h^2+1022056*h^8+59472*h^4)*t^12+h^44*(64*h^30+ 2048303*h^12+2132859*h^16+14224*h^26+234308*h^6+1464932*h^18+672+58160*h^4+1456 *h^28+763966*h^20+1339069*h^10+294390*h^22+9056*h^2+79990*h^24+655838*h^8+ 2385202*h^14)*t^13-h^50*(1+h^2)*(64*h^24+992*h^22+6860*h^20+28494*h^18+79599*h^ 16+156619*h^14+220448*h^12+222607*h^10+160623*h^8+81510*h^6+27896*h^4+5820*h^2+ 560)*t^14+h^56*(41997*h^12+16*h^22+264*h^20+8352*h^16+28581*h^6+280+11358*h^4+ 1952*h^18+53361*h^10+2668*h^2+47163*h^8+22835*h^14)*t^15-h^62*(84+3633*h^6+244* h^14+2043*h^4+2621*h^10+637*h^2+1068*h^12+24*h^16+3917*h^8)*t^16+2*h^68*(3*h^2+ 7)*(2*h^2+1)*(1+h^2)^3*t^17-h^74*(2*h^2+1)*(1+h^2)*t^18)/(t*h^4-1)/(t^2*h^8+1-2 *h^2*t-2*t*h^4)/(1-2*t*h^8-4*t*h^4+6*t^2*h^8+12*t^2*h^10-2*t^3*h^16-6*t*h^6+5*t ^2*h^12-6*t^3*h^14-4*t^3*h^12+t^4*h^16)/(1-8*t*h^4+6*t^2*h^8-4*h^2*t+4*t^2*h^4+ 8*h^6*t^2+8*t^2*h^10-4*t*h^6+4*t^2*h^12-4*t^3*h^14-8*t^3*h^12-4*t^3*h^10+t^4*h^ 16)/(1+516*t^4*h^20-4*t*h^8-20*t*h^4+144*t^2*h^8+33*t^4*h^24-8*h^2*t+24*t^2*h^4 +96*h^6*t^2-20*t^7*h^28+16*t^4*h^8+112*t^2*h^10+144*t^4*h^12-32*t^3*h^6-424*t^5 *h^22+4*t^6*h^32+112*t^6*h^26-4*t^7*h^32+144*t^6*h^24-144*t^5*h^16+96*t^6*h^22+ 24*t^6*h^30-128*t^5*h^26+62*t^6*h^28-32*t^5*h^14+24*t^2*h^14+4*t^2*h^16-20*t^3* h^20-128*t^3*h^18-320*t^3*h^16+64*t^4*h^10-16*t*h^6+62*t^2*h^12-424*t^3*h^14-\ 380*t^3*h^12-280*t^3*h^10+662*t^4*h^16-144*t^3*h^8-8*t^7*h^26-280*t^5*h^18-20*t ^5*h^28+200*t^4*h^22+t^8*h^32-16*t^7*h^30-380*t^5*h^20+24*t^6*h^20+352*t^4*h^14 +752*t^4*h^18-320*t^5*h^24) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /1764901 1624521 1013333 2 3219 3\ n |------- - ------- b + ------- b - ---- b | (1/b) \ 22440 3740 11220 1496 / that in floating-point is: n 66.8056926239638 36.4645094590774 and in Maple input format: HFloat(66.8056926239637647)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 40517 502897 b 107777 b 1289 b /4874 2184 468 2 56 3\ ----- - -------- + --------- - ------- + |---- - ---- b + --- b - --- b | n 5610 5610 5610 2805 \935 187 187 935 / and in Maple-input format: 40517/5610-502897/5610*b+107777/5610*b^2-1289/2805*b^3+(4874/935-2184/187*b+468 /187*b^2-56/935*b^3)*n and in floating point: 4.77835880084157 + 4.89442719140099 n The asymptotic expression for the variance, as a function of n is: 2 3 222736 36273007 b 7765187 b 185698 b ------ - ---------- + ---------- - --------- 8415 42075 42075 42075 /105574 26128 15728 2 622 3\ + |------ - ----- b + ----- b - ----- b | n \42075 2805 8415 14025 / and in Maple-input format: 222736/8415-36273007/42075*b+7765187/42075*b^2-185698/42075*b^3+(105574/42075-\ 26128/2805*b+15728/8415*b^2-622/14025*b^3)*n and in floating point: 2.96536890504092 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 /755497810818251 [1, 3 + ----------------------------------------------- + |--------------- n \ 28160464640 395868535803137 593747659641319 2 28400798598787 3\ / 2 - --------------- b + --------------- b - -------------- b | / n , 15 402292352 2816046464 5632092928 / / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b 25143272 22000363 22000363 176002904 / + ------------------------------------------------------- + | n \ 109528284629096541 286954640515303311 430392155354112057 2 ------------------ - ------------------ b + ------------------ b 275972553472 19712325248 137986276736 20586997557395961 3\ / 2 - ----------------- b | / n , 105 275972553472 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 107998196881527861 1980614777716651269 424378414252057029 2 ------------------ - ------------------- b + ------------------ b 19712325248 9856162624 9856162624 20299343067550257 3\ / 2 - ----------------- b | / n ] 19712325248 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(755497810818251/28160464640-395868535803137/402292352*b+593747659641319/ 2816046464*b^2-28400798598787/5632092928*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 109528284629096541/275972553472-286954640515303311/19712325248*b+ 430392155354112057/137986276736*b^2-20586997557395961/275972553472*b^3)/n^2, 105+(-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752* b^3)/n+(107998196881527861/19712325248-1980614777716651269/9856162624*b+ 424378414252057029/9856162624*b^2-20299343067550257/19712325248*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.761238608831915 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 12.0047124948301 15 - ---------------- + ----------------, n 2 n 106.402545323854 184.604855321300 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 47788542835569 [------------------------------------------------------ + |- -------------- n \ 86241422960 1001579894699661 1502215360621407 2 17963882819019 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 49280813120 344965691840 172482845920 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 119464631934525 5007585548893905 7510619902940235 2 - --------------- + ---------------- b - ---------------- b 2156035574 2464040656 17248284592 89813974807395 3\ / 2 + -------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 210710464120926135 15456404634254124435 473109292840243905 2 - ------------------ + -------------------- b - ------------------ b 34496569184 68993138368 9856162624 9900745700744655 3\ / 2 + ---------------- b | / n ] 8624142296 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-47788542835569/86241422960+1001579894699661/49280813120*b-\ 1502215360621407/344965691840*b^2+17963882819019/172482845920*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-119464631934525/2156035574+5007585548893905/2464040656*b-\ 7510619902940235/17248284592*b^2+89813974807395/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-210710464120926135/34496569184+15456404634254124435/ 68993138368*b-473109292840243905/9856162624*b^2+9900745700744655/8624142296*b^3 )/n^2] and in floating-point it is: 0.00714279180461598 0.0360480831957141 0.714279180461598 3.95954221793470 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 502.750599341107 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 40, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 3, d[1] = 2, d[2] = 2 Then infinity ----- \ n 2 4 4 2 8 ) A(n, h) t = (-(1 + 3 h + h ) (9 h + 9 h + 1) + (1823 h + 5 / ----- n = 0 12 6 2 4 14 16 10 + 789 h + 1208 h + 79 h + 441 h + 207 h + 22 h + 1584 h ) t + ( 18 20 22 2 4 24 12 -8336 h - 2060 h - 284 h - 193 h - 10 - 1501 h - 16 h - 40909 h 14 6 8 10 16 2 2 - 35668 h - 6479 h - 17669 h - 32388 h - 21182 h ) t + (226 h 16 26 30 22 6 14 + 541374 h + 11390 h + 116 h + 144137 h + 12909 h + 464091 h 4 18 10 24 28 8 + 2217 h + 10 + 471896 h + 143036 h + 48805 h + 1668 h + 50714 h 20 12 3 28 12 8 + 304953 h + 298063 h ) t + (-576399 h - 699218 h - 60260 h 4 22 14 18 24 2 - 1549 h - 3975690 h - 1613630 h - 4115079 h - 2700403 h - 130 h 20 6 10 34 30 - 4569659 h - 11513 h - 234794 h - 4736 h - 173806 h 16 36 32 26 4 10 - 2909256 h - 276 h - 5 - 36625 h - 1424276 h ) t + (165723 h 26 24 6 36 18 + 22414273 h + 25789913 h + 4524 h + 366895 h + 11027671 h 12 4 8 20 42 28 + 670583 h + 465 h + 31596 h + 18116629 h + 296 h + 15790372 h 14 16 34 32 30 + 1 + 2133985 h + 5412454 h + 1424867 h + 4068240 h + 8981626 h 2 40 38 22 5 2 16 + 31 h + 6614 h + 64079 h + 24036685 h ) t - h (11517241 h 6 46 2 10 4 40 + 6473 h + 160 h + 36 h + 270211 h + 611 h + 436648 h 38 18 8 14 36 + 2084984 h + 26282028 h + 48252 h + 4104843 h + 7141696 h 28 34 42 32 + 87956707 h + 18575551 h + 60346 h + 1 + 38199266 h 24 44 26 20 12 + 95694675 h + 4788 h + 100657187 h + 49085171 h + 1180032 h 30 22 6 6 14 28 + 63814755 h + 75408509 h ) t + h (8 + 15511383 h + 245920814 h 2 38 6 40 20 + 258 h + 7405083 h + 37433 h + 1745368 h + 147603168 h 30 8 22 24 12 + 182628324 h + 250732 h + 216762956 h + 267626604 h + 4935492 h 26 10 42 4 18 + 278995021 h + 1258991 h + 288511 h + 3928 h + 84037054 h 34 48 44 16 46 + 57484716 h + 32 h + 30380 h + 39758490 h + 1704 h 32 36 7 10 14 + 113129005 h + 23407408 h ) t - h (28 + 38070471 h 28 2 38 6 40 + 497272904 h + 874 h + 17388604 h + 116461 h + 4414040 h 20 30 8 22 + 308299822 h + 374956186 h + 741207 h + 440323848 h 24 12 26 10 42 + 536630022 h + 12934141 h + 559189597 h + 3513939 h + 813811 h 4 18 34 48 44 + 12787 h + 183205573 h + 123710205 h + 224 h + 100916 h 16 46 32 36 8 14 + 91604846 h + 7324 h + 237058972 h + 52226986 h ) t + h (56 14 28 2 38 6 + 62145086 h + 731022920 h + 1716 h + 27799270 h + 217648 h 40 20 30 8 22 + 7365278 h + 457875239 h + 555920452 h + 1344917 h + 644427252 h 24 12 26 10 42 + 781653279 h + 21919574 h + 816623723 h + 6171064 h + 1442845 h 4 18 34 48 44 + 24536 h + 278938622 h + 187988547 h + 608 h + 195298 h 16 46 32 36 9 18 + 144115090 h + 16158 h + 355152952 h + 81022631 h ) t - h (70 14 28 2 38 6 + 68263607 h + 781140853 h + 2106 h + 30661322 h + 256403 h 40 20 30 8 22 + 8253795 h + 490640575 h + 595812955 h + 1552621 h + 688296724 h 24 12 26 10 42 + 833687486 h + 24419281 h + 871158301 h + 6990931 h + 1653437 h 4 18 34 48 44 + 29511 h + 300545982 h + 203474243 h + 808 h + 230898 h 16 46 32 36 10 22 + 156552160 h + 19991 h + 382248774 h + 88379175 h ) t + h (56 14 28 2 38 6 + 50712571 h + 606882386 h + 1646 h + 23401253 h + 192375 h 40 20 30 8 22 + 6231676 h + 380359239 h + 461418505 h + 1149232 h + 537537211 h 24 12 26 10 42 + 652000946 h + 17981188 h + 679494394 h + 5140769 h + 1230219 h 4 18 34 48 44 + 22562 h + 230000916 h + 156940613 h + 550 h + 168642 h 16 46 32 36 11 26 + 117952718 h + 14232 h + 295417528 h + 67917979 h ) t - h (28 14 28 2 38 6 + 25463317 h + 343177465 h + 798 h + 12248530 h + 89768 h 40 20 30 8 22 + 3146266 h + 211457488 h + 258266320 h + 535697 h + 304224670 h 24 12 26 10 42 + 371781801 h + 8717101 h + 386987596 h + 2428002 h + 589097 h 4 18 34 48 44 + 10677 h + 124339354 h + 85661646 h + 184 h + 74727 h 16 46 32 36 12 30 + 61539876 h + 5627 h + 163429497 h + 36433922 h ) t + h (8 14 28 2 38 6 + 8552151 h + 140814945 h + 219 h + 4305222 h + 24307 h 40 20 30 8 22 + 1034673 h + 82904016 h + 104140769 h + 149849 h + 122956370 h 24 12 26 10 42 + 152966541 h + 2741494 h + 159985028 h + 716186 h + 175443 h 4 18 34 48 44 + 2873 h + 46749799 h + 32716211 h + 24 h + 19124 h 16 46 32 36 13 34 + 21962569 h + 1132 h + 64304431 h + 13422287 h ) t - h ( 32 46 2 6 4 28 8 17791600 h + 92 h + 26 h + 3258 h + 350 h + 41436201 h + 22899 h 42 16 12 40 22 + 30261 h + 5203914 h + 538854 h + 210115 h + 1 + 34880871 h 10 44 18 20 36 + 125039 h + 2548 h + 11918125 h + 22453284 h + 3311356 h 34 30 38 26 14 + 8603583 h + 29936227 h + 978200 h + 47361498 h + 1859493 h 24 14 42 30 40 36 4 + 44725552 h ) t + h (1509473 h + 128 h + 24154 h + 1566 h + 6 2 24 38 16 18 34 + 140 h + 8574107 h + 2628 h + 4086568 h + 6776589 h + 134917 h 14 28 20 12 22 + 1988300 h + 3384967 h + 9079488 h + 777536 h + 9825094 h 8 26 32 6 10 15 50 + 59789 h + 6019737 h + 522650 h + 11334 h + 242749 h ) t - h ( 10 20 8 12 24 28 203133 h + 1210224 h + 66751 h + 475766 h + 424151 h + 48733 h 6 18 22 34 4 16 + 16385 h + 1388753 h + 817150 h + 76 h + 2826 h + 1241909 h 14 32 30 26 2 16 58 + 868767 h + 1276 h + 9973 h + 167110 h + 303 h + 15) t + h 2 26 24 22 20 18 16 (1 + h ) (16 h + 264 h + 1944 h + 8453 h + 24030 h + 46820 h 14 12 10 8 6 4 + 64373 h + 63715 h + 45784 h + 23785 h + 8737 h + 2145 h 2 17 66 2 18 16 14 12 + 312 h + 20) t - h (1 + h ) (24 h + 243 h + 1032 h + 2483 h 10 8 6 4 2 18 + 3741 h + 3595 h + 2194 h + 827 h + 173 h + 15) t 74 2 10 8 6 4 2 19 + h (1 + h ) (12 h + 67 h + 125 h + 105 h + 42 h + 6) t 82 2 2 20 / 4 2 8 4 6 - h (2 h + 1) (1 + h ) t ) / ((t h - 1) (t h + 1 - 2 t h - t h ) / 2 8 4 6 8 4 2 8 2 10 (t h + 1 - 2 t h - 2 t h ) (1 - 2 t h - 4 t h + 6 t h + 12 t h 3 16 6 2 12 3 14 3 12 4 16 4 16 - 2 t h - 6 t h + 5 t h - 6 t h - 4 t h + t h ) (t h 3 14 3 12 2 12 3 10 2 10 3 8 2 8 - 2 t h - 8 t h + t h - 9 t h + 4 t h - 2 t h + 6 t h 6 2 6 2 4 4 2 2 2 2 + 16 h t - 2 t h + 20 t h - 8 t h + 8 t h - 9 h t + t - 2 t + 1) 4 20 8 4 2 8 4 24 2 (1 - 4 t + 772 t h - 4 t h - 34 t h + 320 t h + 33 t h - 22 h t 2 4 2 2 6 2 7 28 5 12 4 8 + 221 t h + 60 t h + 376 h t - 34 t h - 290 t h + 676 t h 2 10 4 12 2 3 4 3 6 5 22 + 152 t h + 1092 t h + 6 t - 4 t + t - 796 t h - 722 t h 6 32 3 2 6 26 7 32 6 24 5 16 + 4 t h - 54 t h + 152 t h - 4 t h + 320 t h - 1224 t h 6 22 6 30 5 26 3 4 6 28 5 14 + 376 t h + 24 t h - 148 t h - 290 t h + 62 t h - 796 t h 2 14 2 16 3 20 3 18 3 16 4 10 + 24 t h + 4 t h - 20 t h - 148 t h - 438 t h + 848 t h 4 6 6 2 12 3 14 3 12 3 10 + 352 t h - 20 t h + 62 t h - 722 t h - 922 t h - 1178 t h 4 16 5 10 5 8 7 24 3 8 7 26 + 2074 t h - 54 t h - 4 t h - 4 t h - 1224 t h - 22 t h 5 18 5 28 4 22 8 32 7 30 5 20 - 1178 t h - 20 t h + 232 t h + t h - 20 t h - 922 t h 4 2 4 4 6 16 6 18 6 20 4 14 + 16 t h + 104 t h + 6 t h + 60 t h + 221 t h + 1720 t h 4 18 5 24 + 1584 t h - 438 t h )) and in Maple input format: (-(1+3*h^2+h^4)*(9*h^4+9*h^2+1)+(1823*h^8+5+789*h^12+1208*h^6+79*h^2+441*h^4+ 207*h^14+22*h^16+1584*h^10)*t+(-8336*h^18-2060*h^20-284*h^22-193*h^2-10-1501*h^ 4-16*h^24-40909*h^12-35668*h^14-6479*h^6-17669*h^8-32388*h^10-21182*h^16)*t^2+( 226*h^2+541374*h^16+11390*h^26+116*h^30+144137*h^22+12909*h^6+464091*h^14+2217* h^4+10+471896*h^18+143036*h^10+48805*h^24+1668*h^28+50714*h^8+304953*h^20+ 298063*h^12)*t^3+(-576399*h^28-699218*h^12-60260*h^8-1549*h^4-3975690*h^22-\ 1613630*h^14-4115079*h^18-2700403*h^24-130*h^2-4569659*h^20-11513*h^6-234794*h^ 10-4736*h^34-173806*h^30-2909256*h^16-276*h^36-5-36625*h^32-1424276*h^26)*t^4+( 165723*h^10+22414273*h^26+25789913*h^24+4524*h^6+366895*h^36+11027671*h^18+ 670583*h^12+465*h^4+31596*h^8+18116629*h^20+296*h^42+15790372*h^28+1+2133985*h^ 14+5412454*h^16+1424867*h^34+4068240*h^32+8981626*h^30+31*h^2+6614*h^40+64079*h ^38+24036685*h^22)*t^5-h^2*(11517241*h^16+6473*h^6+160*h^46+36*h^2+270211*h^10+ 611*h^4+436648*h^40+2084984*h^38+26282028*h^18+48252*h^8+4104843*h^14+7141696*h ^36+87956707*h^28+18575551*h^34+60346*h^42+1+38199266*h^32+95694675*h^24+4788*h ^44+100657187*h^26+49085171*h^20+1180032*h^12+63814755*h^30+75408509*h^22)*t^6+ h^6*(8+15511383*h^14+245920814*h^28+258*h^2+7405083*h^38+37433*h^6+1745368*h^40 +147603168*h^20+182628324*h^30+250732*h^8+216762956*h^22+267626604*h^24+4935492 *h^12+278995021*h^26+1258991*h^10+288511*h^42+3928*h^4+84037054*h^18+57484716*h ^34+32*h^48+30380*h^44+39758490*h^16+1704*h^46+113129005*h^32+23407408*h^36)*t^ 7-h^10*(28+38070471*h^14+497272904*h^28+874*h^2+17388604*h^38+116461*h^6+ 4414040*h^40+308299822*h^20+374956186*h^30+741207*h^8+440323848*h^22+536630022* h^24+12934141*h^12+559189597*h^26+3513939*h^10+813811*h^42+12787*h^4+183205573* h^18+123710205*h^34+224*h^48+100916*h^44+91604846*h^16+7324*h^46+237058972*h^32 +52226986*h^36)*t^8+h^14*(56+62145086*h^14+731022920*h^28+1716*h^2+27799270*h^ 38+217648*h^6+7365278*h^40+457875239*h^20+555920452*h^30+1344917*h^8+644427252* h^22+781653279*h^24+21919574*h^12+816623723*h^26+6171064*h^10+1442845*h^42+ 24536*h^4+278938622*h^18+187988547*h^34+608*h^48+195298*h^44+144115090*h^16+ 16158*h^46+355152952*h^32+81022631*h^36)*t^9-h^18*(70+68263607*h^14+781140853*h ^28+2106*h^2+30661322*h^38+256403*h^6+8253795*h^40+490640575*h^20+595812955*h^ 30+1552621*h^8+688296724*h^22+833687486*h^24+24419281*h^12+871158301*h^26+ 6990931*h^10+1653437*h^42+29511*h^4+300545982*h^18+203474243*h^34+808*h^48+ 230898*h^44+156552160*h^16+19991*h^46+382248774*h^32+88379175*h^36)*t^10+h^22*( 56+50712571*h^14+606882386*h^28+1646*h^2+23401253*h^38+192375*h^6+6231676*h^40+ 380359239*h^20+461418505*h^30+1149232*h^8+537537211*h^22+652000946*h^24+ 17981188*h^12+679494394*h^26+5140769*h^10+1230219*h^42+22562*h^4+230000916*h^18 +156940613*h^34+550*h^48+168642*h^44+117952718*h^16+14232*h^46+295417528*h^32+ 67917979*h^36)*t^11-h^26*(28+25463317*h^14+343177465*h^28+798*h^2+12248530*h^38 +89768*h^6+3146266*h^40+211457488*h^20+258266320*h^30+535697*h^8+304224670*h^22 +371781801*h^24+8717101*h^12+386987596*h^26+2428002*h^10+589097*h^42+10677*h^4+ 124339354*h^18+85661646*h^34+184*h^48+74727*h^44+61539876*h^16+5627*h^46+ 163429497*h^32+36433922*h^36)*t^12+h^30*(8+8552151*h^14+140814945*h^28+219*h^2+ 4305222*h^38+24307*h^6+1034673*h^40+82904016*h^20+104140769*h^30+149849*h^8+ 122956370*h^22+152966541*h^24+2741494*h^12+159985028*h^26+716186*h^10+175443*h^ 42+2873*h^4+46749799*h^18+32716211*h^34+24*h^48+19124*h^44+21962569*h^16+1132*h ^46+64304431*h^32+13422287*h^36)*t^13-h^34*(17791600*h^32+92*h^46+26*h^2+3258*h ^6+350*h^4+41436201*h^28+22899*h^8+30261*h^42+5203914*h^16+538854*h^12+210115*h ^40+1+34880871*h^22+125039*h^10+2548*h^44+11918125*h^18+22453284*h^20+3311356*h ^36+8603583*h^34+29936227*h^30+978200*h^38+47361498*h^26+1859493*h^14+44725552* h^24)*t^14+h^42*(1509473*h^30+128*h^40+24154*h^36+1566*h^4+6+140*h^2+8574107*h^ 24+2628*h^38+4086568*h^16+6776589*h^18+134917*h^34+1988300*h^14+3384967*h^28+ 9079488*h^20+777536*h^12+9825094*h^22+59789*h^8+6019737*h^26+522650*h^32+11334* h^6+242749*h^10)*t^15-h^50*(203133*h^10+1210224*h^20+66751*h^8+475766*h^12+ 424151*h^24+48733*h^28+16385*h^6+1388753*h^18+817150*h^22+76*h^34+2826*h^4+ 1241909*h^16+868767*h^14+1276*h^32+9973*h^30+167110*h^26+303*h^2+15)*t^16+h^58* (1+h^2)*(16*h^26+264*h^24+1944*h^22+8453*h^20+24030*h^18+46820*h^16+64373*h^14+ 63715*h^12+45784*h^10+23785*h^8+8737*h^6+2145*h^4+312*h^2+20)*t^17-h^66*(1+h^2) *(24*h^18+243*h^16+1032*h^14+2483*h^12+3741*h^10+3595*h^8+2194*h^6+827*h^4+173* h^2+15)*t^18+h^74*(1+h^2)*(12*h^10+67*h^8+125*h^6+105*h^4+42*h^2+6)*t^19-h^82*( 2*h^2+1)*(1+h^2)*t^20)/(t*h^4-1)/(t^2*h^8+1-2*t*h^4-t*h^6)/(t^2*h^8+1-2*t*h^4-2 *t*h^6)/(1-2*t*h^8-4*t*h^4+6*t^2*h^8+12*t^2*h^10-2*t^3*h^16-6*t*h^6+5*t^2*h^12-\ 6*t^3*h^14-4*t^3*h^12+t^4*h^16)/(t^4*h^16-2*t^3*h^14-8*t^3*h^12+t^2*h^12-9*t^3* h^10+4*t^2*h^10-2*t^3*h^8+6*t^2*h^8+16*h^6*t^2-2*t*h^6+20*t^2*h^4-8*t*h^4+8*t^2 *h^2-9*h^2*t+t^2-2*t+1)/(1-4*t+772*t^4*h^20-4*t*h^8-34*t*h^4+320*t^2*h^8+33*t^4 *h^24-22*h^2*t+221*t^2*h^4+60*t^2*h^2+376*h^6*t^2-34*t^7*h^28-290*t^5*h^12+676* t^4*h^8+152*t^2*h^10+1092*t^4*h^12+6*t^2-4*t^3+t^4-796*t^3*h^6-722*t^5*h^22+4*t ^6*h^32-54*t^3*h^2+152*t^6*h^26-4*t^7*h^32+320*t^6*h^24-1224*t^5*h^16+376*t^6*h ^22+24*t^6*h^30-148*t^5*h^26-290*t^3*h^4+62*t^6*h^28-796*t^5*h^14+24*t^2*h^14+4 *t^2*h^16-20*t^3*h^20-148*t^3*h^18-438*t^3*h^16+848*t^4*h^10+352*t^4*h^6-20*t*h ^6+62*t^2*h^12-722*t^3*h^14-922*t^3*h^12-1178*t^3*h^10+2074*t^4*h^16-54*t^5*h^ 10-4*t^5*h^8-4*t^7*h^24-1224*t^3*h^8-22*t^7*h^26-1178*t^5*h^18-20*t^5*h^28+232* t^4*h^22+t^8*h^32-20*t^7*h^30-922*t^5*h^20+16*t^4*h^2+104*t^4*h^4+6*t^6*h^16+60 *t^6*h^18+221*t^6*h^20+1720*t^4*h^14+1584*t^4*h^18-438*t^5*h^24) We now present a statistical analysis Let, b, be the algebraic number, 4 3 2 RootOf(_Z - 72 _Z + 338 _Z - 72 _Z + 1, index = 1) 4 3 2 whose minimal polynomial is, b - 72 b + 338 b - 72 b + 1 and whose floating-point appx. , to, 10, digits is, 0.0149322967114543 We have the following proven facts The total number of tilings of the region is asymptotic to: /4215091 21121393 4511863 2 62677 3\ n |------- - -------- b + ------- b - ----- b | (1/b) \ 46200 46200 46200 46200 / that in floating-point is: n 84.4308638498834 66.9689344729479 and in Maple input format: HFloat(84.4308638498833801)*HFloat(66.9689344729478790)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 16273 46573 b 52067 b 92 b /20549 1941 2665 2 31 3\ ----- - ------- + -------- - ----- + |----- - ---- b + ---- b - --- b | n 3080 2310 9240 1155 \4620 154 924 770 / and in Maple-input format: 16273/3080-46573/2310*b+52067/9240*b^2-92/1155*b^3+(20549/4620-1941/154*b+2665/ 924*b^2-31/770*b^3)*n and in floating point: 4.98364065875333 + 4.26027334754384 n The asymptotic expression for the variance, as a function of n is: 2 3 406831 b 1219 b 4393 93529 b / 292 521 2 3 4789\ - --------- + ------- + ---- - ------- + |- --- b + --- b - 4/175 b + ----| n 34650 6930 1386 34650 \ 35 315 1575/ and in Maple-input format: -406831/34650*b^2+1219/6930*b^3+4393/1386-93529/34650*b+(-292/35*b+521/315*b^2-\ 4/175*b^3+4789/1575)*n and in floating point: 3.12662929303916 + 2.91642561768438 n The even alpha coefficients until the, 8, -th , are: 10071977 1860345 2663935 2 1359 3 - -------- - ------- b + -------- b - ------- b 31565996 7891499 31565996 1127357 / [1, 3 + ------------------------------------------------- + | n \ 4229721857689 31087513402069 282029554414063 2 2098744183369 3\ - ------------- + -------------- b - --------------- b + ------------- b | 267363986120 26736398612 53472797224 26736398612 / / 2 / n , 15 / 127691880753 13381073505 24298021815 2 12501351 3 - ------------ - ----------- b + ----------- b - --------- b 26736398612 6684099653 26736398612 954871379 / + -------------------------------------------------------------- + | n \ 10811342765266389 198721211446211427 3611459671931089383 2 - ----------------- + ------------------ b - ------------------- b 45291459248728 11322864812182 45291459248728 13437546113629557 3\ / 2 + ----------------- b | / n , 105 11322864812182 / / 127419293721 6252927570 14750749455 2 15473214 3 - ------------ - ---------- b + ----------- b - --------- b 1909742758 954871379 1909742758 136410197 / + ------------------------------------------------------------- + | n \ 10861484120411385 199919450902128189 3639700061518595631 2 - ----------------- + ------------------ b - ------------------- b 3235104232052 808776058013 3235104232052 13542702463821039 3\ / 2 + ----------------- b | / n ] 808776058013 / / and in Maple-input format: [1, 3+(-10071977/31565996-1860345/7891499*b+2663935/31565996*b^2-1359/1127357*b ^3)/n+(-4229721857689/267363986120+31087513402069/26736398612*b-282029554414063 /53472797224*b^2+2098744183369/26736398612*b^3)/n^2, 15+(-127691880753/ 26736398612-13381073505/6684099653*b+24298021815/26736398612*b^2-12501351/ 954871379*b^3)/n+(-10811342765266389/45291459248728+198721211446211427/ 11322864812182*b-3611459671931089383/45291459248728*b^2+13437546113629557/ 11322864812182*b^3)/n^2, 105+(-127419293721/1909742758-6252927570/954871379*b+ 14750749455/1909742758*b^2-15473214/136410197*b^3)/n+(-10861484120411385/ 3235104232052+199919450902128189/808776058013*b-3639700061518595631/ 3235104232052*b^2+13542702463821039/808776058013*b^3)/n^2] and in floating-point it is: 0.322578132263974 0.366549734211393 [1, 3 - ----------------- + -----------------, n 2 n 4.80564740984938 5.58676672715327 15 - ---------------- + ----------------, n 2 n 66.8167197023482 82.8924981567179 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 34073379 1025460972 238681809 2 2382372 3 ----------- + ---------- b - ---------- b + ---------- b 33420498265 6684099653 6684099653 4774356895 / 10244587213269 [---------------------------------------------------------- + |- -------------- n \ 56614324060910 346580913584418 3550140778981761 2 26428615884978 3\ / 2 + --------------- b - ---------------- b + -------------- b | / n , 28307162030455 56614324060910 28307162030455 / / 681467580 102546097200 23868180900 2 47647440 3 ---------- + ------------ b - ----------- b + --------- b 6684099653 6684099653 6684099653 954871379 / ----------------------------------------------------------- + | n \ 101221026247350 6821684616903285 5068274354397630 2 - --------------- + ---------------- b - ---------------- b 5661432406091 5661432406091 808776058013 528246105074085 3\ / 2 + --------------- b | / n , 5661432406091 / / 1533302055 230728718700 53703407025 2 107206740 3 ---------- + ------------ b - ----------- b + --------- b 136410197 136410197 136410197 19487171 / ----------------------------------------------------------- + | n \ 4954024363336695 2318243311667394765 12283384700649461265 2 - ---------------- + ------------------- b - -------------------- b 2541867610898 17793073276286 17793073276286 182902348381198365 3\ / 2 + ------------------ b | / n ] 17793073276286 / / and in Maple-input format: [(34073379/33420498265+1025460972/6684099653*b-238681809/6684099653*b^2+2382372 /4774356895*b^3)/n+(-10244587213269/56614324060910+346580913584418/ 28307162030455*b-3550140778981761/56614324060910*b^2+26428615884978/ 28307162030455*b^3)/n^2, (681467580/6684099653+102546097200/6684099653*b-\ 23868180900/6684099653*b^2+47647440/954871379*b^3)/n+(-101221026247350/ 5661432406091+6821684616903285/5661432406091*b-5068274354397630/808776058013*b^ 2+528246105074085/5661432406091*b^3)/n^2, (1533302055/136410197+230728718700/ 136410197*b-53703407025/136410197*b^2+107206740/19487171*b^3)/n+(-\ 4954024363336695/2541867610898+2318243311667394765/17793073276286*b-\ 12283384700649461265/17793073276286*b^2+182902348381198365/17793073276286*b^3)/ n^2] and in floating-point it is: 0.00330245744902289 0.0121082504372843 0.330245744902289 1.28350631922617 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 36.4095933740273 157.349640550966 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. ------------------------------------------------------------------- Theorem Number, 41, : Let A(n,h) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + 2 n + c[2], by , d[1] + 2 n + d[2], rectangle the central , 2 n, by , 2 n, square. where , c[1] = 2, c[2] = 3, d[1] = 3, d[2] = 1 Then infinity ----- \ n 2 4 4 2 2 ) A(n, h) t = (-(1 + 3 h + h ) (9 h + 9 h + 1) + h / ----- n = 0 12 8 6 10 2 14 4 (290 h + 1482 h + 1369 h + 901 h + 160 h + 38 h + 12 + 683 h ) t 4 2 18 16 14 12 10 - h (1 + h ) (52 h + 560 h + 2669 h + 7300 h + 12448 h 8 6 4 2 2 6 2 + 13569 h + 9348 h + 3838 h + 812 h + 60) t + h (2520 h 14 4 8 10 22 26 + 214903 h + 16028 h + 136379 h + 222031 h + 3376 h + 24 h 6 20 12 18 24 16 + 57982 h + 16722 h + 257445 h + 55553 h + 416 h + 129240 h 3 8 30 14 18 12 + 160) t - h (32 h + 1582719 h + 934994 h + 1358216 h 16 26 2 22 8 28 + 1397020 h + 7476 h + 4160 h + 172514 h + 413108 h + 732 h 24 4 6 10 20 4 + 44413 h + 31224 h + 138832 h + 240 + 873735 h + 469510 h ) t + 10 2 30 28 26 24 22 h (1 + h ) (384 h + 6430 h + 49794 h + 232583 h + 727295 h 20 18 16 14 12 + 1615142 h + 2650771 h + 3302566 h + 3167760 h + 2340365 h 10 8 6 4 2 5 12 + 1314526 h + 546780 h + 161632 h + 31824 h + 3712 h + 192) t - h 2 32 30 28 26 24 (1 + h ) (1952 h + 29369 h + 202930 h + 853612 h + 2445678 h 22 20 18 16 14 + 5068414 h + 7889083 h + 9460802 h + 8886233 h + 6582857 h 12 10 8 6 4 2 + 3828954 h + 1718192 h + 575924 h + 136992 h + 21280 h + 1856 h 6 16 8 2 22 30 + 64) t + h (384 + 1018536 h + 6976 h + 3876692 h + 5496 h 4 10 6 16 12 + 57888 h + 2612628 h + 292784 h + 9756369 h + 5153346 h 20 28 14 18 24 + 6932564 h + 72581 h + 7982091 h + 9353738 h + 1573249 h 26 2 2 7 20 2 32 30 + 434542 h ) (1 + h ) t - h (1 + h ) (9414 h + 129667 h 28 26 24 22 20 + 816518 h + 3137134 h + 8284514 h + 16054287 h + 23769980 h 18 16 14 12 10 + 27586166 h + 25486087 h + 18889635 h + 11243718 h + 5343380 h 8 6 4 2 8 24 2 + 1996288 h + 567792 h + 115488 h + 14848 h + 896) t + h (1 + h ) ( 32 30 28 26 24 10154 h + 138677 h + 866686 h + 3308791 h + 8693377 h 22 20 18 16 14 + 16782126 h + 24784507 h + 28725766 h + 26529668 h + 19670011 h 12 10 8 6 4 + 11721814 h + 5584692 h + 2096240 h + 601008 h + 123936 h 2 9 28 2 32 30 28 + 16320 h + 1024) t - h (1 + h ) (6900 h + 95724 h + 608090 h 26 24 22 20 18 + 2361200 h + 6309404 h + 12370246 h + 18497082 h + 21611586 h 16 14 12 10 8 + 20029533 h + 14847323 h + 8814306 h + 4156940 h + 1524820 h 6 4 2 10 32 2 32 + 418768 h + 80928 h + 9856 h + 576) t + h (1 + h ) (2856 h 30 28 26 24 22 + 41568 h + 277171 h + 1128145 h + 3146061 h + 6390401 h 20 18 16 14 12 + 9812498 h + 11671801 h + 10926750 h + 8110263 h + 4758518 h 10 8 6 4 2 11 + 2174140 h + 752120 h + 188432 h + 31776 h + 3136 h + 128) t - 38 10 18 14 20 22 h (2590180 h + 8125516 h + 7446592 h + 5917899 h + 3327213 h 26 24 28 30 16 + 448 + 429953 h + 1406554 h + 89384 h + 11296 h + 8753727 h 32 6 12 2 8 4 12 + 656 h + 295896 h + 4983921 h + 7472 h + 1022056 h + 59472 h ) t 44 30 12 16 26 6 + h (64 h + 2048303 h + 2132859 h + 14224 h + 234308 h 18 4 28 20 10 + 1464932 h + 672 + 58160 h + 1456 h + 763966 h + 1339069 h 22 2 24 8 14 13 50 + 294390 h + 9056 h + 79990 h + 655838 h + 2385202 h ) t - h 2 24 22 20 18 16 14 (1 + h ) (64 h + 992 h + 6860 h + 28494 h + 79599 h + 156619 h 12 10 8 6 4 2 + 220448 h + 222607 h + 160623 h + 81510 h + 27896 h + 5820 h 14 56 12 22 20 16 6 + 560) t + h (41997 h + 16 h + 264 h + 8352 h + 28581 h + 280 4 18 10 2 8 14 15 + 11358 h + 1952 h + 53361 h + 2668 h + 47163 h + 22835 h ) t - 62 6 14 4 10 2 12 h (84 + 3633 h + 244 h + 2043 h + 2621 h + 637 h + 1068 h 16 8 16 68 2 2 2 3 17 + 24 h + 3917 h ) t + 2 h (3 h + 7) (2 h + 1) (1 + h ) t 74 2 2 18 / 4 - h (2 h + 1) (1 + h ) t ) / ((t h - 1) / 2 8 2 4 8 4 2 8 2 10 (t h + 1 - 2 h t - 2 t h ) (1 - 2 t h - 4 t h + 6 t h + 12 t h 3 16 6 2 12 3 14 3 12 4 16 4 - 2 t h - 6 t h + 5 t h - 6 t h - 4 t h + t h ) (1 - 8 t h 2 8 2 2 4 6 2 2 10 6 2 12 + 6 t h - 4 h t + 4 t h + 8 h t + 8 t h - 4 t h + 4 t h 3 14 3 12 3 10 4 16 4 20 8 - 4 t h - 8 t h - 4 t h + t h ) (1 + 516 t h - 4 t h 4 2 8 4 24 2 2 4 6 2 - 20 t h + 144 t h + 33 t h - 8 h t + 24 t h + 96 h t 7 28 4 8 2 10 4 12 3 6 5 22 - 20 t h + 16 t h + 112 t h + 144 t h - 32 t h - 424 t h 6 32 6 26 7 32 6 24 5 16 6 22 + 4 t h + 112 t h - 4 t h + 144 t h - 144 t h + 96 t h 6 30 5 26 6 28 5 14 2 14 2 16 + 24 t h - 128 t h + 62 t h - 32 t h + 24 t h + 4 t h 3 20 3 18 3 16 4 10 6 2 12 - 20 t h - 128 t h - 320 t h + 64 t h - 16 t h + 62 t h 3 14 3 12 3 10 4 16 3 8 7 26 - 424 t h - 380 t h - 280 t h + 662 t h - 144 t h - 8 t h 5 18 5 28 4 22 8 32 7 30 5 20 - 280 t h - 20 t h + 200 t h + t h - 16 t h - 380 t h 6 20 4 14 4 18 5 24 + 24 t h + 352 t h + 752 t h - 320 t h )) and in Maple input format: (-(1+3*h^2+h^4)*(9*h^4+9*h^2+1)+h^2*(290*h^12+1482*h^8+1369*h^6+901*h^10+160*h^ 2+38*h^14+12+683*h^4)*t-h^4*(1+h^2)*(52*h^18+560*h^16+2669*h^14+7300*h^12+12448 *h^10+13569*h^8+9348*h^6+3838*h^4+812*h^2+60)*t^2+h^6*(2520*h^2+214903*h^14+ 16028*h^4+136379*h^8+222031*h^10+3376*h^22+24*h^26+57982*h^6+16722*h^20+257445* h^12+55553*h^18+416*h^24+129240*h^16+160)*t^3-h^8*(32*h^30+1582719*h^14+934994* h^18+1358216*h^12+1397020*h^16+7476*h^26+4160*h^2+172514*h^22+413108*h^8+732*h^ 28+44413*h^24+31224*h^4+138832*h^6+240+873735*h^10+469510*h^20)*t^4+h^10*(1+h^2 )*(384*h^30+6430*h^28+49794*h^26+232583*h^24+727295*h^22+1615142*h^20+2650771*h ^18+3302566*h^16+3167760*h^14+2340365*h^12+1314526*h^10+546780*h^8+161632*h^6+ 31824*h^4+3712*h^2+192)*t^5-h^12*(1+h^2)*(1952*h^32+29369*h^30+202930*h^28+ 853612*h^26+2445678*h^24+5068414*h^22+7889083*h^20+9460802*h^18+8886233*h^16+ 6582857*h^14+3828954*h^12+1718192*h^10+575924*h^8+136992*h^6+21280*h^4+1856*h^2 +64)*t^6+h^16*(384+1018536*h^8+6976*h^2+3876692*h^22+5496*h^30+57888*h^4+ 2612628*h^10+292784*h^6+9756369*h^16+5153346*h^12+6932564*h^20+72581*h^28+ 7982091*h^14+9353738*h^18+1573249*h^24+434542*h^26)*(1+h^2)^2*t^7-h^20*(1+h^2)* (9414*h^32+129667*h^30+816518*h^28+3137134*h^26+8284514*h^24+16054287*h^22+ 23769980*h^20+27586166*h^18+25486087*h^16+18889635*h^14+11243718*h^12+5343380*h ^10+1996288*h^8+567792*h^6+115488*h^4+14848*h^2+896)*t^8+h^24*(1+h^2)*(10154*h^ 32+138677*h^30+866686*h^28+3308791*h^26+8693377*h^24+16782126*h^22+24784507*h^ 20+28725766*h^18+26529668*h^16+19670011*h^14+11721814*h^12+5584692*h^10+2096240 *h^8+601008*h^6+123936*h^4+16320*h^2+1024)*t^9-h^28*(1+h^2)*(6900*h^32+95724*h^ 30+608090*h^28+2361200*h^26+6309404*h^24+12370246*h^22+18497082*h^20+21611586*h ^18+20029533*h^16+14847323*h^14+8814306*h^12+4156940*h^10+1524820*h^8+418768*h^ 6+80928*h^4+9856*h^2+576)*t^10+h^32*(1+h^2)*(2856*h^32+41568*h^30+277171*h^28+ 1128145*h^26+3146061*h^24+6390401*h^22+9812498*h^20+11671801*h^18+10926750*h^16 +8110263*h^14+4758518*h^12+2174140*h^10+752120*h^8+188432*h^6+31776*h^4+3136*h^ 2+128)*t^11-h^38*(2590180*h^10+8125516*h^18+7446592*h^14+5917899*h^20+3327213*h ^22+448+429953*h^26+1406554*h^24+89384*h^28+11296*h^30+8753727*h^16+656*h^32+ 295896*h^6+4983921*h^12+7472*h^2+1022056*h^8+59472*h^4)*t^12+h^44*(64*h^30+ 2048303*h^12+2132859*h^16+14224*h^26+234308*h^6+1464932*h^18+672+58160*h^4+1456 *h^28+763966*h^20+1339069*h^10+294390*h^22+9056*h^2+79990*h^24+655838*h^8+ 2385202*h^14)*t^13-h^50*(1+h^2)*(64*h^24+992*h^22+6860*h^20+28494*h^18+79599*h^ 16+156619*h^14+220448*h^12+222607*h^10+160623*h^8+81510*h^6+27896*h^4+5820*h^2+ 560)*t^14+h^56*(41997*h^12+16*h^22+264*h^20+8352*h^16+28581*h^6+280+11358*h^4+ 1952*h^18+53361*h^10+2668*h^2+47163*h^8+22835*h^14)*t^15-h^62*(84+3633*h^6+244* h^14+2043*h^4+2621*h^10+637*h^2+1068*h^12+24*h^16+3917*h^8)*t^16+2*h^68*(3*h^2+ 7)*(2*h^2+1)*(1+h^2)^3*t^17-h^74*(2*h^2+1)*(1+h^2)*t^18)/(t*h^4-1)/(t^2*h^8+1-2 *h^2*t-2*t*h^4)/(1-2*t*h^8-4*t*h^4+6*t^2*h^8+12*t^2*h^10-2*t^3*h^16-6*t*h^6+5*t ^2*h^12-6*t^3*h^14-4*t^3*h^12+t^4*h^16)/(1-8*t*h^4+6*t^2*h^8-4*h^2*t+4*t^2*h^4+ 8*h^6*t^2+8*t^2*h^10-4*t*h^6+4*t^2*h^12-4*t^3*h^14-8*t^3*h^12-4*t^3*h^10+t^4*h^ 16)/(1+516*t^4*h^20-4*t*h^8-20*t*h^4+144*t^2*h^8+33*t^4*h^24-8*h^2*t+24*t^2*h^4 +96*h^6*t^2-20*t^7*h^28+16*t^4*h^8+112*t^2*h^10+144*t^4*h^12-32*t^3*h^6-424*t^5 *h^22+4*t^6*h^32+112*t^6*h^26-4*t^7*h^32+144*t^6*h^24-144*t^5*h^16+96*t^6*h^22+ 24*t^6*h^30-128*t^5*h^26+62*t^6*h^28-32*t^5*h^14+24*t^2*h^14+4*t^2*h^16-20*t^3* h^20-128*t^3*h^18-320*t^3*h^16+64*t^4*h^10-16*t*h^6+62*t^2*h^12-424*t^3*h^14-\ 380*t^3*h^12-280*t^3*h^10+662*t^4*h^16-144*t^3*h^8-8*t^7*h^26-280*t^5*h^18-20*t ^5*h^28+200*t^4*h^22+t^8*h^32-16*t^7*h^30-380*t^5*h^20+24*t^6*h^20+352*t^4*h^14 +752*t^4*h^18-320*t^5*h^24) We now present a statistical analysis Let, b, be the algebraic number, 2 4 3 RootOf(1 - 42 _Z + 203 _Z + _Z - 42 _Z , index = 1) 2 4 3 whose minimal polynomial is, 1 - 42 b + 203 b + b - 42 b and whose floating-point appx. , to, 10, digits is, 0.0274239257523061 We have the following proven facts The total number of tilings of the region is asymptotic to: /1764901 1624521 1013333 2 3219 3\ n |------- - ------- b + ------- b - ---- b | (1/b) \ 22440 3740 11220 1496 / that in floating-point is: n 66.8056926239638 36.4645094590774 and in Maple input format: HFloat(66.8056926239637647)*HFloat(36.4645094590773624)^n The asymptotic expression for the expected number of horizontal tiles as a function of n, is: 2 3 40517 502897 b 107777 b 1289 b /4874 2184 468 2 56 3\ ----- - -------- + --------- - ------- + |---- - ---- b + --- b - --- b | n 5610 5610 5610 2805 \935 187 187 935 / and in Maple-input format: 40517/5610-502897/5610*b+107777/5610*b^2-1289/2805*b^3+(4874/935-2184/187*b+468 /187*b^2-56/935*b^3)*n and in floating point: 4.77835880084157 + 4.89442719140099 n The asymptotic expression for the variance, as a function of n is: 2 3 7765187 b 222736 185698 b 36273007 b ---------- + ------ - --------- - ---------- 42075 8415 42075 42075 / 26128 15728 2 622 3 105574\ + |- ----- b + ----- b - ----- b + ------| n \ 2805 8415 14025 42075 / and in Maple-input format: 7765187/42075*b^2+222736/8415-185698/42075*b^3-36273007/42075*b+(-26128/2805*b+ 15728/8415*b^2-622/14025*b^3+105574/42075)*n and in floating point: 2.96536890504092 + 2.25514247021766 n The even alpha coefficients until the, 8, -th , are: 225223 9776715 2194855 2 422601 3 - ------- - ------- b + ------- b - ------- b 1026256 897974 897974 7183792 /755497810818251 [1, 3 + ----------------------------------------------- + |--------------- n \ 28160464640 395868535803137 593747659641319 2 28400798598787 3\ / 2 - --------------- b + --------------- b - -------------- b | / n , 15 402292352 2816046464 5632092928 / / 87213963 3392617815 762289455 2 146787381 3 - -------- - ---------- b + --------- b - --------- b 25143272 22000363 22000363 176002904 / + ------------------------------------------------------- + | n \ 109528284629096541 286954640515303311 430392155354112057 2 ------------------ - ------------------ b + ------------------ b 275972553472 19712325248 137986276736 20586997557395961 3\ / 2 - ----------------- b | / n , 105 275972553472 / / 16665177 580416885 130539945 2 25139799 3 - -------- - --------- b + --------- b - -------- b 326536 285719 285719 2285752 / + ----------------------------------------------------- + | n \ 107998196881527861 1980614777716651269 424378414252057029 2 ------------------ - ------------------- b + ------------------ b 19712325248 9856162624 9856162624 20299343067550257 3\ / 2 - ----------------- b | / n ] 19712325248 / / and in Maple-input format: [1, 3+(-225223/1026256-9776715/897974*b+2194855/897974*b^2-422601/7183792*b^3)/ n+(755497810818251/28160464640-395868535803137/402292352*b+593747659641319/ 2816046464*b^2-28400798598787/5632092928*b^3)/n^2, 15+(-87213963/25143272-\ 3392617815/22000363*b+762289455/22000363*b^2-146787381/176002904*b^3)/n+( 109528284629096541/275972553472-286954640515303311/19712325248*b+ 430392155354112057/137986276736*b^2-20586997557395961/275972553472*b^3)/n^2, 105+(-16665177/326536-580416885/285719*b+130539945/285719*b^2-25139799/2285752* b^3)/n+(107998196881527861/19712325248-1980614777716651269/9856162624*b+ 424378414252057029/9856162624*b^2-20299343067550257/19712325248*b^3)/n^2] and in floating-point it is: 0.516202509697840 0.761238608831915 [1, 3 - ----------------- + -----------------, n 2 n 7.67160972762144 12.0047124948301 15 - ---------------- + ----------------, n 2 n 106.402545323854 184.604855321300 105 - ---------------- + ----------------] n 2 n Note that, at least until the, 8, -th moment as n goes to infinity they tend to the even moments of the Normal distribution. (Recall that the 2r-th moment of N(0,1) is: (2r)!/(2^r*r!) ) The SQUARES of the odd alpha coefficients until the, 8, -th , are: 8889021 80129979 17727903 2 17036973 3 - --------- + -------- b - -------- b + ---------- b 502865440 88001452 88001452 3520058080 / 47788542835569 [------------------------------------------------------ + |- -------------- n \ 86241422960 1001579894699661 1502215360621407 2 17963882819019 3\ / 2 + ---------------- b - ---------------- b + -------------- b | / n , 49280813120 344965691840 172482845920 / / 44445105 2003249475 443197575 2 85184865 3 - -------- + ---------- b - --------- b + --------- b 25143272 22000363 22000363 176002904 / ------------------------------------------------------- + | n \ 119464631934525 5007585548893905 7510619902940235 2 - --------------- + ---------------- b - ---------------- b 2156035574 2464040656 17248284592 89813974807395 3\ / 2 + -------------- b | / n , 8624142296 / / 400005945 18029245275 3988778175 2 766663785 3 - --------- + ----------- b - ---------- b + --------- b 2052512 1795948 1795948 14367584 / ---------------------------------------------------------- + | n \ 210710464120926135 15456404634254124435 473109292840243905 2 - ------------------ + -------------------- b - ------------------ b 34496569184 68993138368 9856162624 9900745700744655 3\ / 2 + ---------------- b | / n ] 8624142296 / / and in Maple-input format: [(-8889021/502865440+80129979/88001452*b-17727903/88001452*b^2+17036973/ 3520058080*b^3)/n+(-47788542835569/86241422960+1001579894699661/49280813120*b-\ 1502215360621407/344965691840*b^2+17963882819019/172482845920*b^3)/n^2, (-\ 44445105/25143272+2003249475/22000363*b-443197575/22000363*b^2+85184865/ 176002904*b^3)/n+(-119464631934525/2156035574+5007585548893905/2464040656*b-\ 7510619902940235/17248284592*b^2+89813974807395/8624142296*b^3)/n^2, (-\ 400005945/2052512+18029245275/1795948*b-3988778175/1795948*b^2+766663785/ 14367584*b^3)/n+(-210710464120926135/34496569184+15456404634254124435/ 68993138368*b-473109292840243905/9856162624*b^2+9900745700744655/8624142296*b^3 )/n^2] and in floating-point it is: 0.00714279180461598 0.0360480831957141 0.714279180461598 3.95954221793470 [------------------- - ------------------, ----------------- - ----------------, n 2 n 2 n n 78.7492796441412 502.750599341107 ---------------- - ----------------] n 2 n Note that, at least until the, 8, -th moment they tend to ZERO. (Recall that all the odd moments of N(0,1) are 0.) So at least as far as the, 8, -th moment our random variable appears to be asymptotically normal This ends this theorem. -------------------------------------------------------------------