{[x,0],[0,1], x is a power of 4} Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([4^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [0., 2.676, 0., 3.907, 0., 4.911, 0., 5.487, 0., 6.326] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([4^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [0., 7.907000000000000000000000000000000000000, 0., 15.86100000000000000000000000000000000000, 0., 23.83000000000000000000000000000000000000, 0., 31.52900000000000000000000000000000000000, 0., 39.43300000000000000000000000000000000000] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(60, 60, {seq([4^i, 0], i = 1 .. 50), [0, 1]}, 1000); 0.6697227777777777777777777777777777777778 (ix) average proportion of good walks (1000 trials) SimRepeatSteps(60, 60, {seq([4^i, 0], i = 1 .. 50), [0, 1]}, 1000); 0.5408929468599033816425120772946859903382 (x) OEIS number - N/A {[1,2],[2,1]} Chess Knight Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {[1, 2], [2, 1]}, 1000), i = 1 .. 10)] = [0., 0., 4.720000000000000000000000000000000000000, 0., 0., 7.142000000000000000000000000000000000000, 0., 0., 8.598000000000000000000000000000000000000, 0.] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{[1,2],[2,1]} ,1000) = 0 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{[1,2],[2,1]} ,1000) = FAIL (x) OEIS number - N/A {[x,0],[0,1], x is a power of 3} Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (vi) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (vi) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([3^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [0., 0., 3.854, 0., 0., 5.373, 0., 0., 6.709, 0.] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([3^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [0., 0., 14.23600000000000000000000000000000000000, 0., 0., 28.36200000000000000000000000000000000000, 0., 0., 42.13500000000000000000000000000000000000, 0.] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(60, 60, {seq([3^i, 0], i = 1 .. 50), [0, 1]}, 1000); 0.6256890356956146429830640356956146429831 (ix) average proportion of good walks (1000 trials) SimGoodWalks(60, 60, {seq([3^i, 0], i = 1 .. 50), [0, 1]}, 1000); 0.5366438395885764306816938395885764306817 (x) OEIS number - N/A {[1, 2], [2, 1], [2, 2]} Number of walks sequence data: (i) recurrence operator (n + 2)/(n + 4) - 2*(2*n + 5)*N/(n + 4) - 2*(n + 3)*N^2/(n + 4) + N^4 (ii) growth constant 1.946965328128404560260818238630271227735 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k 1.521097460*10^216*4.334450859^n*(1. - 5441.555921/n + 1.484840950*10^7/n^2)/n^114.8237815 (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator (n + 1)/(n + 7) - 2*(2*n + 5)*N/(n + 7) - 2*(n + 4)*N^2/(n + 7) + N^4 (ii) growth constant 1.946965328128404560260818238630271227735 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k 3.369786030*10^(-45)*1.664099496^n*n^21.86311625*(1. + 1153.993107/n + 656306.5528/(n^2)) (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [2.723, 4.185, 5.385, 6.263, 7.279, 7.83, 8.595, 9.277, 9.991, 10.47] (viii) average number of repeat steps (1000 trials) 0.3160846053094106486764996219946164328812 (ix) average proportion of good walks (1000 trials) 0.6387464580654792000731933991110854292167 (x) OEIS number - A182883, A25250 {[x,0],[0,1], x is a power of 2} Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([2^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [2.549, 3.551, 4.554, 5.319, 6.013, 6.68, 7.07, 7.864, 8.43, 8.526] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([2^i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [5.957, 11.694,17.588, 23.379,29.047, 34.761, 40.763, 46.543, 52.491,58.162] (viii) average number of repeat steps (1000 trials) 0.5453868261150391867301944983957947798739 (ix) average proportion of good walks (1000 trials) 0.5357530277243977246932783866091788534110 (x) OEIS number - N/A {[x,0],[0,1], x is a prime number} Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([ithprime(i), 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [2.414, 3.61, 4.467, 5.269, 5.79, 6.305, 7.056, 7.559, 8.128, 8.745] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([ithprime(i), 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [5.848, 11.522, 17.139, 22.942, 28.911, 34.293, 40.13,46.056,51.682, 57.179] (viii) average number of repeat steps (1000 trials) 0.5397693619649309039861636962937958185008 (ix) average proportion of good walks (1000 trials) 0.5415449659406466749762292459818015509224 (x) OEIS number - N/A {[1,1]} 1D reduction Number of walks sequence data: (i) recurrence operator N-1 (ii) growth constant 1 (iii) critical exponent 0 (iv) estimated asymptotics to order k k=2 1.0*1.^n (v) congruence L[ithprime(i)] mod ithprime(i)^p = 1 for all p Number of Good walks sequence data: (i) recurrence operator N - 1 (ii) growth constant 1 (iii) critical exponent 0 (iv) estimated asymptotics to order k k=2 1.0*1.^n (v) congruence L[ithprime(i)] mod ithprime(i)^p = 1 for all p (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {[1, 1]}, 1000), i = 1 .. 10)] = [10., 20., 30., 40., 50., 60., 70., 80., 90., 100.] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{[1,2],[2,1]} ,1000) = 0.9800000000000000000000000000000000000000 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{[1,2],[2,1]} ,1000) = 1 (x) OEIS number - A12 {[1,1],[2,2]} Fibonacci Number of walks sequence data: (i) recurrence operator N^2-N-1 (ii) growth constant 1.618033988749894848204586834365638117720 (iii) critical exponent 0 (iv) estimated asymptotics to order k .7236067978*1.618033989^n*(1.-0.1017882578e-10/n+0.8173556077e-10/n^2)/n^0.2112108836e-12 (v) congruence Number of Good walks sequence data: (i) recurrence operator N^2-N-1 (ii) growth constant 1.618033988749894848204586834365638117720 (iii) critical exponent 0 (iv) estimated asymptotics to order k .7236067978*1.618033989^n*(1.-0.1017882578e-10/n+0.8173556077e-10/n^2)/n^0.2112108836e-12 (v) congruence (vi) average visits to diagonal (1000 trials) [7.37, 14.591, 21.849, 29.069, 36.371, 43.606, 50.912, 58.091, 65.428, 72.402] (viii) average number of repeat steps (1000 trials) 0.5162503872616816490999565198228886653447 (ix) average proportion of good walks (1000 trials) 1 (x) OEIS number - A45 {[1,1],...,[i,i], i = 1..n} Chess Bishop Number of walks sequence data: (i) recurrence operator N-2 (ii) growth constant 2 (iii) critical exponent 0 (iv) estimated asymptotics to order k k=2 .5000000000*2.000000000^n (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator N-2 (ii) growth constant 2 (iii) critical exponent 0 (iv) estimated asymptotics to order k k=2 .5000000000*2.000000000^n (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..15 (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {[1, 1]}, 1000), i = 1 .. 10)] = [10., 20., 30., 40., 50., 60., 70., 80., 90., 100.] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{seq([i, i], i = 1 .. 20)} ,1000) = 0.3163753802486176878103056230411208669608 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{seq([i, i], i = 1 .. 20)} ,1000) = 1 (x) OEIS number - A79 {[1, 1], [1, 2], [2, 1], [2, 2]} Number of walks sequence data: (i) recurrence operator (n+2)/(n+4)-(2*n+5)*N/(n+4)-(n+3)*N^2/(n+4)-(2*n+7)*N^3/(n+4)+N^4 (ii) growth constant 2.618033988749894848204586834365638117720 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .5000000000*2.000000000^n (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator .4938952382*2.618034036^n*(1.-.1927812843/n-0.8661653101e-3/n^2)/n^.5000034372 (n+1)/(n+7)-(2*n+5)*N/(n+7)-(n+4)*N^2/(n+7)-(2*n+11)*N^3/(n+7)+N^4 (ii) growth constant 2.618033988749894848204586834365638117720 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 2.878869114*2.618021765^n*(1.-3.855234012/n+11.73179838/n^2)/n^1.499090179 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..15 (vi) average visits to diagonal (1000 trials) [4.15, 6.063, 7.786, 8.887, 10.037, 10.969, 11.875, 12.69, 13.82, 14.728] (viii) average number of repeat steps (1000 trials) 0.2650133353463457103115790724338636443144 (ix) average proportion of good walks (1000 trials) 0.6264944549324301835062236783901059367343 (x) OEIS number - A51286, A4148 {[1, 1], [1, 2], [2, 0], [2, 1], [2, 2]} Number of walks sequence data: (i) recurrence operator -(3/2)*(16*n+49)*(n+2)/((n+4)*(16*n+33))+(1/2)*(16*n^2+65*n+58)*N/((n+4)*(16*n+33))-(1/2)*(48*n^2+211*n+220)*N^2/((n+4)*(16*n+33))-(1/2)*(80*n^2+453*n+604)*N^3/((n+4)*(16*n+33))+N^4 (ii) growth constant 3. (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .5000000000*2.000000000^n (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator .3664367422*2.999999862^n*(1.-.2181217659/n-0.6373517744e-1/n^2)/n^.4999912063 -3*(n+1)/(n+4)-(2*n+5)*N/(n+4)+N^2 (ii) growth constant 3. (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 1.463951765*2.999995845^n*(1.-2.417840820/n+4.832210410/n^2)/n^1.499731426 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 (vi) average visits to diagonal (1000 trials) [3.299, 4.643, 5.94, 6.846, 7.561, 8.553, 9.1220, 9.922, 10.353, 11.18] (viii) average number of repeat steps (1000 trials) 0.2429655802293709288014309183895924762973 (ix) average proportion of good walks (1000 trials) 0.5957947173029726515640091746117044256123 (x) OEIS number - A1006 {[0, 1], [1, 1], [2, 1]} Number of walks sequence data: (i) recurrence operator -3*(n+1)/(n+2)-(2*n+3)*N/(n+2)+N^2 (ii) growth constant 3. (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .4886065688*3.000000028^n*(1.-.1876258580/n+0.3944418280e-2/n^2)/n^.5000017681 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 L[ithprime(i)] mod ithprime(i)^2 = 1 for i = 3..95 Number of Good walks sequence data: (i) recurrence operator -3*(n+1)/(n+4)-(2*n+5)*N/(n+4)+N^2 (ii) growth constant 3. (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 1.463951765*2.999995845^n*(1.-2.417840820/n+4.832210410/n^2)/n^1.499731426 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {[0, 1], [1, 1], [2, 1]}, 1000), i = 1 .. 10)]; [3.898, 6.116, 7.552, 8.723, 10.054, 10.685, 11.95, 12.801, 13.637, 14.626] (viii) average number of repeat steps (1000 trials) 0.3217600000000000000000000000000000000000 (ix) average proportion of good walks (1000 trials) 0.5943200000000000000000000000000000000000 (x) OEIS number - A2426, A1006 {[0, 2], [1, 1], [2, 0]} Number of walks sequence data: (i) recurrence operator -3*(n+1)/(n+2)-(2*n+3)*N/(n+2)+N^2 (ii) growth constant 3. (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .4886065688*3.000000028^n*(1.-.1876258580/n+0.3944418280e-2/n^2)/n^.5000017681 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 L[ithprime(i)] mod ithprime(i)^2 = 1 for i = 3..95 Number of Good walks sequence data: (i) recurrence operator -3*(n+1)/(n+4)-(2*n+5)*N/(n+4)+N^2 (ii) growth constant 3. (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 1.463951765*2.999995845^n*(1.-2.417840820/n+4.832210410/n^2)/n^1.499731426 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 (vi) average visits to diagonal (1000 trials) [3.972, 6.031, 7.597, 8.914, 9.959, 10.964, 11.633, 13.065, 13.551, 14.539] (viii) average number of repeat steps (1000 trials) 0.3205400000000000000000000000000000000000 (ix) average proportion of good walks (1000 trials) 0.6050000000000000000000000000000000000000 (x) OEIS number - A2426, A1006 {[0,1],[1,0]} Standard Number of walks sequence data: (i) recurrence operator -2*(2*n + 1)/(n + 1) + N (ii) growth constant 4 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 0.5641910832*4.000000012^n*(1. - 0.1250406798/n + 0.008468438739/(n^2))/n^0.5000005650 (v) congruence L[ithprime(i)] mod ithprime(i) = 2 for i = 3..95 L[ithprime(i)] mod ithprime(i)^2 = 2 for i = 3..95 L[ithprime(i)] mod ithprime(i)^3 = 2 for i = 3..95 Number of Good walks sequence data: (i) recurrence operator -2*(2*n + 1)/(n + 2) + N (ii) growth constant 4 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 0.5641019615*3.999999317^n*(1. - 1.122606001/n + 1.091159539/(n^2))/n^1.499967030 (v) congruence L[ithprime(i)] mod ithprime(i) = 2 for i = 3..95 (vi) average visits to diagonal (1000 trials) [4.547, 6.94, 8.807, 10.311, 11.485, 12.593,13.356, 14.779,15.965,16.914] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {[0, 1], [1, 0]}, 1000), i = 1 .. 10)]; [9.953, 20.101, 29.839, 39.97, 50.032, 59.971, 69.998, 79.746,90.186, 100.02] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{[1,0],[0,1]},1000) = 0.4889400000000000000000000000000000000000 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{[1,0],[0,1]},1000) = 0.5570800000000000000000000000000000000000 (x) OEIS number - A984, A108 {[x^4,0],[0,1]} x to the 4th power Number of walks sequence data: (i) recurrence operator -2*(2*n + 1)/(n + 1) + N (ii) growth constant 4 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 0.5639760089*4.000129249^n*(1. - 0.09670629009/n - 0.2487200155/(n^2))/n^0.5003554467 (v) congruence Number of Good walks sequence data: (i) recurrence operator -2*(2*n + 1)/(n + 2) + N (ii) growth constant 4 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 0.5638869210*4.000128553^n*(1. - 1.094271611/n + 0.8057056803/(n^2))/n^1.500321912 (v) congruence (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([i^3, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [4.745, 7.035, 8.621, 10.134, 11.707, 12.948, 13.729, 15.124, 15.795, 16.458] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([i^4, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [9.983, 19.987, 29.863, 40.027, 50.217, 59.823, 70.257, 80.103, 89.857, 100.228] (viii) average number of repeat steps (1000 trials) 0.4900500000000000000000000000000000000000 (ix) average proportion of good walks (1000 trials) 0.5638641176470588235294117647058823529412 (x) OEIS number - A000984 {[x^3,0],[0,1]} x cubed Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([i^3, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [4.704, 6.92, 8.421, 9.87, 10.947, 11.696, 12.964, 13.838, 15.191, 15.419] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([i^3, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [10.016, 19.785, 29.729, 39.218, 49.246, 58.661, 69.114, 78.075, 88.159, 98.344] (viii) average number of repeat steps (1000 trials) 0.4890667216804201050262565641410352588147 (ix) average proportion of good walks (1000 trials) 0.5588032058014503625906476619154788697174 (x) OEIS number - N/A {[x^2,0],[0,1]} x squared Number of walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator - N/A (ii) growth constant - N/A (iii) critical exponent - N/A (iv) estimated asymptotics to order k - N/A (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([i^2, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [4.092, 5.942, 7.196, 8.536, 9.404, 10.353, 10.87, 12.025, 12.815, 13.348] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([i^2, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [9.188, 18.27, 27.435, 36.333, 45.389, 54.106, 63.065, 72.339, 80.838, 90.193] (viii) average number of repeat steps (1000 trials) 0.4688796227767119098223016046984666399265 (ix) average proportion of good walks (1000 trials) 0.5387824988165379407061258464102381563527 (x) OEIS number - N/A {[i,0],[0,1], i = 1..n} Number of walks sequence data: (i) recurrence operator n/(n + 1) - 3*(2*n + 3)*N/(n + 2) + N^2 (ii) growth constant 5.828427124746190097603377448419396157140 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 0.4049466223*5.828427118^n*(1. - 0.1477311938/n - 0.001398632584/(n^2))/n^0.4999997661 (v) congruence L[ithprime(i)] mod ithprime(i) = 3 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator n/(n + 3) - 3*(2*n + 3)*N/(n + 3) + N^2 (ii) growth constant 5.828427124746190097603377448419396157140 (iii) critical exponent -3/2 (vi) estimated asymptotics to order k k=2 0.4048826554*5.828426105^n*(1. - 1.145296515/n + 1.103927738/(n^2))/n^1.499966231 (v) congruence (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [3.492, 5.197, 6.363, 7.797, 8.664, 9.338, 10.214, 10.88, 11.536, 12.083] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([i, 0], i = 1 .. 50), [0, 1]}, 1000), i = 1 .. 10)]; [8.352,16.431, 24.96,33.304, 41.468, 49.858, 58.119,66.356,74.452, 83.152] (viii) average number of repeat steps (1000 trials) 0.4257538887305949428614106333085986762656 (ix) average proportion of good walks (1000 trials) 0.5608657518761097417876928458668041253212 (x) OEIS number - A176479, A1003 {[0, 1], [1, 0], [1, 2], [2, 1]} Number of walks sequence data: (i) recurrence operator -2*(2*n+3)/(n+3)-8*(n+2)*N/(n+3)-2*(2*n+5)*N^2/(n+3)+N^3 (ii) growth constant 5.566315770083119384217400910852095578045 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .4939590021*5.566315799^n*(1.-.1878817793/n-0.3584900392e-2/n^2)/n^.5000009817 (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator -2*(2*n+3)/(n+5)-2*(6*n+13)*N/(n+5)-2*(6*n+17)*N^2/(n+5)-(3*n+8)*N^3/(n+5)+N^4 (ii) growth constant 5.566315770083119384217400910852095578045 (iii) critical exponent 1 (iv) estimated asymptotics to order k k=2 .6443409580*5.566315134^n*(1.-1.391811996/n+1.392516456/n^2)/n^1.499977803 (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [4.039, 6.009, 7.397, 8.87, 10.188, 11.058, 12.213, 12.718, 13.64, 15.064] (viii) average number of repeat steps (1000 trials) 0.3615729717457527330965628719055811469401 (ix) average proportion of good walks (1000 trials) 0.5742361433730704836203149196770448771938 (x) OEIS number - A137635, A73155 {[0,i],[i,0], i = 1..n} Chess Rook Number of walks sequence data: (i) recurrence operator 9*n/(n + 2) - 2*(5*n + 7)*N/(n + 2) + N^2 (ii) growth constant 9 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 0.2659605107*8.999999958^n*(1. - 0.1561929778/n - 0.006262333936/(n^2))/n^0.4999991857 (v) congruence L[ithprime(i)] mod ithprime(i) = 2 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator 9*n/(n + 3) - 5*(2*n + 3)*N/(n + 3) + N^2 (ii) growth constant 9 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 0.2991525073*8.999998199^n*(1. - 1.215964877/n + 1.135632174/(n^2))/n^1.499961518 (v) congruence L[ithprime(i)] mod ithprime(i) = 2 for i = 3..15 (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([i, i], i = 1 .. 20)}, 1000), i = 1 .. 10)]; [5.533, 10.552, 15.469, 20.64, 25.412, 30.44,35.637,40.325,45.152, 50.338] (vii) average direction change (1000 trials) [seq(SimDirectionChange(10*i, 10*i, {seq([0, i], i = 1 .. 20), seq([i, 0], i = 1 .. 20)}, 1000), i = 1 .. 10)]; [6.828, 13.49, 20.234, 27.086, 33.55, 40.178, 46.616, 53.74, 60.275, 66.51] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{seq([0, i], i = 1 .. 20), seq([i, 0], i = 1 .. 20)} ,1000) = 0.2458846360958470651005865375338495850977 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{seq([0, i], i = 1 .. 20), seq([i, 0], i = 1 .. 20)} ,1000) = 0.5368221560324001687298680135638937332394 (x) OEIS number - A51708, A59231 {[0,1],[1,0],[1,1]} Chess King going in NE direction Number of walks sequence data: (i) recurrence operator (n + 1)/(n + 2) - 3*(2*n + 3)*N/(n + 2) + N^2 (ii) growth constant 5.828427124746190097603377448419396157140 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 0.5726839165*5.828427151^n*(1. - 0.1174783484/n + 0.01188971354/(n^2))/n^0.5000008481 (v) congruence L[ithprime(i)] mod ithprime(i) = 3 for i = 3..95 Number of Good walks sequence data: (i) recurrence operator n/(n + 3) - 3*(2*n + 3)*N/(n + 3) + N^2 (ii) growth constant 5.828427124746190097603377448419396157140 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 0.8097653107*5.828426105^n*(1. - 1.145296515/n + 1.103927738/(n^2))/n^1.499966231 (v) congruence L[ithprime(i)] mod ithprime(i) = 4 for i = 3..95 (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {[0, 1], [1, 0], [1, 1]}, 1000), i = 1 .. 10)] = [4.803, 7.072, 8.97, 10.3, 11.209, 13.158, 14.237, 15.247, 16.544, 17.237] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{[1,0],[0,1],[1,1]},1000) = 0.3630215718121088523376646516440498695908 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{[1,0],[0,1],[1,1]},1000) = 0.5628211678321937466236401420570264456763 (x) OEIS number - A1850, A6318 {[0, 1], [1, 0], [1, 1], [2, 2]} Number of walks sequence data: (i) recurrence operator (n+2)/(n+4)+(2*n+5)*N/(n+4)-(n+3)*N^2/(n+4)-3*(2*n+7)*N^3/(n+4)+N^4 (ii) growth constant 6.105739095572730853522172896607904689109 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .5622042924*6.105739155^n*(1.-.1241745278/n+0.1456760503e-1/n^2)/n^.5000018458 (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator (n-1)/(n+5)+(2*n+1)*N/(n+5)-(n+2)*N^2/(n+5)-3*(2*n+7)*N^3/(n+5)+N^4 (ii) growth constant 6.105739095572730853522172896607904689109 (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 .8640864483*6.105737841^n*(1.-1.125014018/n+1.063651131/n^2)/n^1.499960479 (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [4.677, 6.969, 8.799, 10.063, 11.372, 12.847, 13.833, 15.158, 15.647, 16.555] (viii) average number of repeat steps (1000 trials) 0.3444074783280696384423756286317269094744 (ix) average proportion of good walks (1000 trials) 0.5679445404405117024646069703472083940692 (x) OEIS number - A191649, A175934 {[0, 1], [1, 0], [1, 1], [1, 2], [2, 1]} Number of walks sequence data: (i) recurrence operator -2*(2*n+3)/(n+3)-7*(n+2)*N/(n+3)-3*(2*n+5)*N^2/(n+3)+N^3 (ii) growth constant 7.070105942718174702750832392351662467997 (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .5232342699*7.070105947^n*(1.-.1624202523/n-0.3452952279e-2/n^2)/n^.5000001173 (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator -2*(2*n+3)/(n+5)-(11*n+23)*N/(n+5)-(13*n+41)*N^2/(n+5)-5*(n+3)*N^3/(n+5)+N^4 (ii) growth constant 7.070105942718174702750832392351662467997 (iii) critical exponent 1 (iv) estimated asymptotics to order k k=2 .8252145122*7.070105114^n*(1.-1.382837329/n+1.375989849/n^2)/n^1.499977244 (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [4.311, 6.493, 7.93, 9.398, 10.779, 11.73, 12.884, 13.921, 14.548, 15.446] (viii) average number of repeat steps (1000 trials) 0.3034308697470514723009298592374791054988 (ix) average proportion of good walks (1000 trials) 0.5684500368607730412218311776115634325401 (x) OEIS number - A339565 (submitted) {[1, 1], [1, 2], [2, 0], [2, 1], [2, 2]} Number of walks sequence data: (i) recurrence operator -(3/2)*(16*n+49)*(n+2)/((n+4)*(16*n+33))+(1/2)*(16*n^2+65*n+58)*N/((n+4)*(16*n+33))-(1/2)*(48*n^2+211*n+220)*N^2/((n+4)*(16*n+33))-(1/2)*(80*n^2+453*n+604)*N^3/((n+4)*(16*n+33))+N^4 (ii) growth constant 3. (iii) critical exponent -1/2 (iv) estimated asymptotics to order k k=2 .3664367422*2.999999862^n*(1.-.2181217659/n-0.6373517744e-1/n^2)/n^.4999912063 (v) congruence - N/A Number of Good walks sequence data: (i) recurrence operator -3*(n+1)/(n+4)-(2*n+5)*N/(n+4)+N^2 (ii) growth constant 3. (iii) critical exponent -3/2 (iv) estimated asymptotics to order k k=2 1.463951765*2.999995845^n*(1.-2.417840820/n+4.832210410/n^2)/n^1.499731426 (v) congruence L[ithprime(i)] mod ithprime(i) = 1 for i = 3..95 (vi) average visits to diagonal (1000 trials) [3.299, 4.643, 5.94, 6.846, 7.561, 8.553, 9.1220, 9.922, 10.353, 11.18] (viii) average number of repeat steps (1000 trials) 0.2429655802293709288014309183895924762973 (ix) average proportion of good walks (1000 trials) 0.5957947173029726515640091746117044256123 (x) OEIS number - A1006 {[0,i],[i,0],[i,i], i = 1..n} Chess Queen Number of walks sequence data: (i) recurrence operator 24*n/(n + 4) - (58*n + 81)*N/(n + 4) + (95*n + 237)*N^2/(2*(n + 4)) - (29*n + 98)*N^3/(2*(n + 4)) + N^4 (ii) growth constant 10.47213595499957939281834733746255247088 (iii) critical exponent -3 (vi) estimated asymptotics to order k k=2 0.2914828127*10.47213617^n*(1. - 0.1045533196/n + 0.02812123578/(n^2))/n^0.5000039362 (v) congruence L[ithprime(i)] mod ithprime(i) = 3 for i = 3..15 Number of Good walks sequence data: (i) recurrence operator 24*n/(n + 5) - (58*n + 83)*N/(n + 5) + 19*(5*n + 13)*N^2/(2*(n + 5)) - (29*n + 106)*N^3/(2*(n + 5)) + N^4 (ii) growth constant 10.47213595499957939281834733746255247088 (iii) critical exponent -3 (vi) estimated asymptotics to order k k=2 0.3803326833*10.47213503^n*(1. - 1.066979449/n + 1.111898822/(n^2))/n^1.499982631 (v) congruence - N/A (vi) average visits to diagonal (1000 trials) [seq(SimVisitsToDiagonal(10*i, 10*i, {seq([0, i], i = 1 .. 50), seq([i, 0], i = 1 .. 50), seq([i, i], i = 1 .. 50)}, 1000), i = 1 .. 10)]; [2.976, 4.155, 5.183, 5.901, 6.7, 7.059, 7.741, 8.266, 8.836, 9.145] (viii) average number of repeat steps (1000 trials) SimRepeatSteps(50,50,{seq([0, i], i = 1 .. 50), seq([i, 0], i = 1 .. 50), seq([i, i], i = 1 .. 50)} ,1000) = 0.2173069218610011645193443311036793710076 (ix) average proportion of good walks (1000 trials) SimGoodWalks(50,50,{seq([0, i], i = 1 .. 50), seq([i, 0], i = 1 .. 50), seq([i, i], i = 1 .. 50)} ,1000) = 0.5506233932142289547985333140921481225617 (x) OEIS number - A132595, A175962