Explicit Expressions for The Number of ways of playing Stanley Solitaire sta\ ring with position [a,0^k,a-j] for all a, and j from 0 to , 10 and k from j+1 to, j + 10 By Shalosh B. Ekhad Theorem Number, [0, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, a], is (a + 2) (2 a + 2)! ------------------ 2 ((a + 2)!) and in Maple notation 1/(a+2)!^2*(a+2)*(2*a+2)! For example if a=, 60, then the number is, 6182127958584855650487080847216336 This agrees with the value obtained by numerical dynamical programming that is, 6182127958584855650487080847216336 Theorem Number, [0, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, a], is 2 9 (a + 1) (2 a)! (a + a + 2/3) (a + 2) (a + 3) ----------------------------------------------- 2 ((a + 3)!) and in Maple notation 9*(a+1)*(2*a)!*(a^2+a+2/3)*(a+2)*(a+3)/(a+3)!^2 For example if a=, 60, then the number is, 13359332790472035567424069630879536 This agrees with the value obtained by numerical dynamical programming that is, 13359332790472035567424069630879536 Theorem Number, [0, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, a], is 2 4 (a + a + 3) (a + 2) (a + 3) (a + 4) (2 a + 2)! ------------------------------------------------- 2 ((a + 4)!) and in Maple notation 4*(a^2+a+3)*(a+2)*(a+3)*(a+4)*(2*a+2)!/(a+4)!^2 For example if a=, 60, then the number is, 22465411420928895087037874150152221 This agrees with the value obtained by numerical dynamical programming that is, 22465411420928895087037874150152221 Theorem Number, [0, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, a], is 4 3 2 25 (a + 2 a + 11 a + 10 a + 24/5) (a + 1) (2 a)! (a + 5) (a + 2) (a + 3) / 2 (a + 4) / ((a + 5)!) / and in Maple notation 25*(a^4+2*a^3+11*a^2+10*a+24/5)*(a+1)*(2*a)!*(a+5)*(a+2)*(a+3)*(a+4)/(a+5)!^2 For example if a=, 60, then the number is, 32732264909773661973987064740013512 This agrees with the value obtained by numerical dynamical programming that is, 32732264909773661973987064740013512 Theorem Number, [0, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, a], is 4 3 2 9 (a + 2 a + 77/3 a + 74/3 a + 40) (a + 6) (a + 5) (2 a + 2)! (a + 2) / 2 (a + 3) (a + 4) / ((a + 6)!) / and in Maple notation 9*(a^4+2*a^3+77/3*a^2+74/3*a+40)*(a+6)*(a+5)*(2*a+2)!*(a+2)*(a+3)*(a+4)/(a+6)!^ 2 For example if a=, 60, then the number is, 43379372231538605412304743870240036 This agrees with the value obtained by numerical dynamical programming that is, 43379372231538605412304743870240036 Theorem Number, [0, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a], is 49 (a + 1) (a + 6) (2 a)! (a + 5) (a + 2) 6 5 4 3 2 (a + 3 a + 55 a + 105 a + 304 a + 252 a + 720/7) (a + 3) (a + 4) / 2 (a + 7) / ((a + 7)!) / and in Maple notation 49*(a+1)*(a+6)*(2*a)!*(a+5)*(a+2)*(a^6+3*a^5+55*a^4+105*a^3+304*a^2+252*a+720/7 )*(a+3)*(a+4)*(a+7)/(a+7)!^2 For example if a=, 60, then the number is, 53708655454146386359926372877176216 This agrees with the value obtained by numerical dynamical programming that is, 53708655454146386359926372877176216 Theorem Number, [0, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a], is 6 5 4 3 2 16 (a + 3 a + 100 a + 195 a + 1159 a + 1062 a + 1260) (a + 2) (a + 3) / 2 (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (2 a + 2)! / ((a + 8)!) / and in Maple notation 16*(a^6+3*a^5+100*a^4+195*a^3+1159*a^2+1062*a+1260)*(a+2)*(a+3)*(a+4)*(a+5)*(a+ 6)*(a+7)*(a+8)*(2*a+2)!/(a+8)!^2 For example if a=, 60, then the number is, 63173270171694239943154336107513666 This agrees with the value obtained by numerical dynamical programming that is, 63173270171694239943154336107513666 Theorem Number, [0, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a], is 8 7 6 5 4 3 2 81 (a + 4 a + 518/3 a + 504 a + 12467/3 a + 7476 a + 47492/3 a + 12176 a / + 4480) (2 a)! / ((a + 1) (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) / 2 (a + 7) (a + 8) (a + 9) (a!) ) and in Maple notation 81*(a^8+4*a^7+518/3*a^6+504*a^5+12467/3*a^4+7476*a^3+47492/3*a^2+12176*a+4480)* (2*a)!/(a+1)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/a!^2 For example if a=, 60, then the number is, 71412514481366332772611722455855736 This agrees with the value obtained by numerical dynamical programming that is, 71412514481366332772611722455855736 Theorem Number, [0, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a], is 8 7 6 5 4 3 2 (25 a + 100 a + 6850 a + 20200 a + 282745 a + 531940 a + 2129580 a / + 1866960 a + 1814400) (2 a + 2)! / ((a + 2) (a + 3) (a + 4) (a + 5) / 2 (a + 6) (a + 7) (a + 8) (a + 9) (a + 10) ((a + 1)!) ) and in Maple notation (25*a^8+100*a^7+6850*a^6+20200*a^5+282745*a^4+531940*a^3+2129580*a^2+1866960*a+ 1814400)*(2*a+2)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10)/(a+1)! ^2 For example if a=, 60, then the number is, 78253164378808372869673821373638900 This agrees with the value obtained by numerical dynamical programming that is, 78253164378808372869673821373638900 Theorem Number, [0, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a], is / 10 9 8 7 6 5 4 121 (2 a)! |a + 5 a + 420 a + 1650 a + 29253 a + 82005 a + 449630 a \ 3 2 3628800\ / + 764500 a + 1335096 a + 966240 a + -------| / ((a + 1) (a + 2) 11 / / (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10) (a + 11) 2 (a!) ) and in Maple notation 121*(2*a)!*(a^10+5*a^9+420*a^8+1650*a^7+29253*a^6+82005*a^5+449630*a^4+764500*a ^3+1335096*a^2+966240*a+3628800/11)/(a+1)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/( a+8)/(a+9)/(a+10)/(a+11)/a!^2 For example if a=, 60, then the number is, 83684102881521808706555591530010856 This agrees with the value obtained by numerical dynamical programming that is, 83684102881521808706555591530010856 Theorem Number, [1, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, a - 1], is 6 GAMMA(2 a) ------------------------- 2 (a + 2) GAMMA(a) (a + 1) and in Maple notation 6*GAMMA(2*a)/(a+2)/GAMMA(a)^2/(a+1) For example if a=, 60, then the number is, 4598276993988735607800308068177440 This agrees with the value obtained by numerical dynamical programming that is, 4598276993988735607800308068177440 Theorem Number, [1, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, a - 1], is 2 12 (a + 1) (2 a - 1)! --------------------------------- (a + 3) (a + 2) (a + 1)! (a - 1)! and in Maple notation 12*(a^2+1)*(2*a-1)!/(a+3)/(a+2)/(a+1)!/(a-1)! For example if a=, 60, then the number is, 8761055796483299959623761562702096 This agrees with the value obtained by numerical dynamical programming that is, 8761055796483299959623761562702096 Theorem Number, [1, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, a - 1], is 2 20 (a + 5) GAMMA(2 a) ----------------------------------------- 2 (a + 4) (a + 3) (a + 2) GAMMA(a) (a + 1) and in Maple notation 20*(a^2+5)*GAMMA(2*a)/(a+4)/(a+3)/(a+2)/GAMMA(a)^2/(a+1) For example if a=, 60, then the number is, 13704355624445595127414112587450125 This agrees with the value obtained by numerical dynamical programming that is, 13704355624445595127414112587450125 Theorem Number, [1, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, a - 1], is 4 2 30 (a + 15 a + 8) (2 a - 1)! ------------------------------------------------- (a + 5) (a + 4) (a + 3) (a + 2) (a + 1)! (a - 1)! and in Maple notation 30*(a^4+15*a^2+8)*(2*a-1)!/(a+5)/(a+4)/(a+3)/(a+2)/(a+1)!/(a-1)! For example if a=, 60, then the number is, 19027909285328066846572952152563387 This agrees with the value obtained by numerical dynamical programming that is, 19027909285328066846572952152563387 Theorem Number, [1, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a - 1], is 4 2 42 (a + 35 a + 84) GAMMA(2 a) --------------------------------------------------------- 2 (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) GAMMA(a) (a + 1) and in Maple notation 42*(a^4+35*a^2+84)*GAMMA(2*a)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/GAMMA(a)^2/(a+1) For example if a=, 60, then the number is, 24351462946210538565731791717676649 This agrees with the value obtained by numerical dynamical programming that is, 24351462946210538565731791717676649 Theorem Number, [1, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 1], is 6 4 2 56 (a + 70 a + 469 a + 180) (2 a - 1)! ----------------------------------------------------------------- (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 1)! (a - 1)! and in Maple notation 56*(a^6+70*a^4+469*a^2+180)*(2*a-1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+1)! /(a-1)! For example if a=, 60, then the number is, 29357192507935847794194581159499567 This agrees with the value obtained by numerical dynamical programming that is, 29357192507935847794194581159499567 Theorem Number, [1, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 1], is 6 4 2 72 (a + 126 a + 1869 a + 3044) GAMMA(2 a) ------------------------------------------------------------------------- 2 (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) GAMMA(a) (a + 1) and in Maple notation 72*(a^6+126*a^4+1869*a^2+3044)*GAMMA(2*a)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/( a+8)/GAMMA(a)^2/(a+1) For example if a=, 60, then the number is, 33816077663758392148959754948014099 This agrees with the value obtained by numerical dynamical programming that is, 33816077663758392148959754948014099 Theorem Number, [1, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 1], is 8 6 4 2 90 (a + 210 a + 5985 a + 26060 a + 8064) (2 a - 1)!/((a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 1)! (a - 1)!) and in Maple notation 90*(a^8+210*a^6+5985*a^4+26060*a^2+8064)*(2*a-1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6) /(a+7)/(a+8)/(a+9)/(a+1)!/(a-1)! For example if a=, 60, then the number is, 37596436817607940623651967507841637 This agrees with the value obtained by numerical dynamical programming that is, 37596436817607940623651967507841637 Theorem Number, [1, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 1], is 8 6 4 2 / 110 (a + 330 a + 16401 a + 152900 a + 193248) GAMMA(2 a) / ((a + 2) / 2 (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10) GAMMA(a) (a + 1)) and in Maple notation 110*(a^8+330*a^6+16401*a^4+152900*a^2+193248)*GAMMA(2*a)/(a+2)/(a+3)/(a+4)/(a+5 )/(a+6)/(a+7)/(a+8)/(a+9)/(a+10)/GAMMA(a)^2/(a+1) For example if a=, 60, then the number is, 40656727561200432246021853865797263 This agrees with the value obtained by numerical dynamical programming that is, 40656727561200432246021853865797263 Theorem Number, [1, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 1], is 10 8 6 4 2 132 (a + 495 a + 39963 a + 696905 a + 2286636 a + 604800) (2 a - 1)!/( (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10) (a + 11) (a + 1)! (a - 1)!) and in Maple notation 132*(a^10+495*a^8+39963*a^6+696905*a^4+2286636*a^2+604800)*(2*a-1)!/(a+2)/(a+3) /(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10)/(a+11)/(a+1)!/(a-1)! For example if a=, 60, then the number is, 43027375320321376460533737664213593 This agrees with the value obtained by numerical dynamical programming that is, 43027375320321376460533737664213593 Theorem Number, [2, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, a - 2], is 3 2 (2 a + 4 a - 2 a - 4) (2 a)! ------------------------------ 2 ((a + 2)!) and in Maple notation (2*a^3+4*a^2-2*a-4)*(2*a)!/(a+2)!^2 For example if a=, 60, then the number is, 3014426029392615565113535289138544 This agrees with the value obtained by numerical dynamical programming that is, 3014426029392615565113535289138544 Theorem Number, [2, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, a - 2], is 6 5 4 2 15 a (a + 4 a + 2 a + 9 a - 4 a - 12) (2 a - 2)! ---------------------------------------------------- 2 ((a + 3)!) and in Maple notation 15*a*(a^6+4*a^5+2*a^4+9*a^2-4*a-12)*(2*a-2)!/(a+3)!^2 For example if a=, 60, then the number is, 5340684771963107408779582830196440 This agrees with the value obtained by numerical dynamical programming that is, 5340684771963107408779582830196440 Theorem Number, [2, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, a - 2], is 7 6 5 4 3 2 (6 a + 48 a + 144 a + 372 a + 954 a + 732 a - 1104 a - 1152) (2 a)! ------------------------------------------------------------------------- 2 ((a + 4)!) and in Maple notation (6*a^7+48*a^6+144*a^5+372*a^4+954*a^3+732*a^2-1104*a-1152)*(2*a)!/(a+4)!^2 For example if a=, 60, then the number is, 7957725857354910732903886313886573 This agrees with the value obtained by numerical dynamical programming that is, 7957725857354910732903886313886573 Theorem Number, [2, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a - 2], is 4 3 2 35 (a - 2 a + 23 a - 22 a + 24) a (a + 2) (a + 3) (a + 4) (a + 5) (2 a - 2)! 2 / 2 (a - 1) / ((a + 5)!) / and in Maple notation 35*(a^4-2*a^3+23*a^2-22*a+24)*a*(a+2)*(a+3)*(a+4)*(a+5)*(2*a-2)!/(a+5)!^2*(a^2-\ 1) For example if a=, 60, then the number is, 10664238432845579127938422395309702 This agrees with the value obtained by numerical dynamical programming that is, 10664238432845579127938422395309702 Theorem Number, [2, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 2], is 4 3 2 12 (a - 2 a + 49 a - 48 a + 180) (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) 2 / 2 (2 a)! (a - 1) / ((a + 6)!) / and in Maple notation 12*(a^4-2*a^3+49*a^2-48*a+180)/(a+6)!^2*(a+2)*(a+3)*(a+4)*(a+5)*(a+6)*(2*a)!*(a ^2-1) For example if a=, 60, then the number is, 13281279518237382452062725878999835 This agrees with the value obtained by numerical dynamical programming that is, 13281279518237382452062725878999835 Theorem Number, [2, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 2], is 2 4 3 2 63 (a - a + 10) (a - 2 a + 83 a - 82 a + 72) a (a + 2) (a + 3) (a + 4) 2 / 2 (a + 5) (a + 6) (a + 7) (2 a - 2)! (a - 1) / ((a + 7)!) / and in Maple notation 63*(a^2-a+10)*(a^4-2*a^3+83*a^2-82*a+72)/(a+7)!^2*a*(a+2)*(a+3)*(a+4)*(a+5)*(a+ 6)*(a+7)*(2*a-2)!*(a^2-1) For example if a=, 60, then the number is, 15669967994570888356401211837132620 This agrees with the value obtained by numerical dynamical programming that is, 15669967994570888356401211837132620 Theorem Number, [2, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 2], is 6 5 4 3 2 20 (a - 3 a + 163 a - 321 a + 3364 a - 3204 a + 8064) (a + 2) (a + 3) 2 / 2 (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (2 a)! (a - 1) / ((a + 8)!) / and in Maple notation 20*(a^6-3*a^5+163*a^4-321*a^3+3364*a^2-3204*a+8064)/(a+8)!^2*(a+2)*(a+3)*(a+4)* (a+5)*(a+6)*(a+7)*(a+8)*(2*a)!*(a^2-1) For example if a=, 60, then the number is, 17740164674059926806827899667514367 This agrees with the value obtained by numerical dynamical programming that is, 17740164674059926806827899667514367 Theorem Number, [2, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 2], is 8 7 6 5 4 3 2 99 (2 a - 2)! (a - 4 a + 266 a - 784 a + 10409 a - 19516 a + 69964 a - 60336 a + 40320)/((a - 2)! (a + 1)! (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 99*(2*a-2)!*(a^8-4*a^7+266*a^6-784*a^5+10409*a^4-19516*a^3+69964*a^2-60336*a+ 40320)/(a-2)!/(a+1)!/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 19450327148420436831093424396960158 This agrees with the value obtained by numerical dynamical programming that is, 19450327148420436831093424396960158 Theorem Number, [2, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 2], is 8 7 6 5 4 3 2 30 (a - 4 a + 406 a - 1204 a + 26173 a - 50344 a + 336300 a - 311328 a / 2 + 604800) (a - 1) (2 a)! / ((a!) (a + 1) (a + 2) (a + 3) (a + 4) / (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 30*(a^8-4*a^7+406*a^6-1204*a^5+26173*a^4-50344*a^3+336300*a^2-311328*a+604800)* (a-1)/a!^2/(a+1)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10)*(2*a)! For example if a=, 60, then the number is, 20800455417652418429197786025469993 This agrees with the value obtained by numerical dynamical programming that is, 20800455417652418429197786025469993 Theorem Number, [2, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 2], is 10 9 8 7 6 5 143 (2 a - 2)! (a - 5 a + 600 a - 2370 a + 62013 a - 177765 a 4 3 2 + 1491650 a - 2689780 a + 7517736 a - 6202080 a + 3628800)/((a - 2)! (a + 1)! (a + 11) (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 143*(2*a-2)!*(a^10-5*a^9+600*a^8-2370*a^7+62013*a^6-177765*a^5+1491650*a^4-\ 2689780*a^3+7517736*a^2-6202080*a+3628800)/(a-2)!/(a+1)!/(a+11)/(a+10)/(a+9)/(a +8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 21820974907541381045605308195376488 This agrees with the value obtained by numerical dynamical programming that is, 21820974907541381045605308195376488 Theorem Number, [3, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, a - 3], is 5 3 (5 a - 25 a + 20 a) (2 a - 2)! -------------------------------- 2 ((a + 2)!) and in Maple notation (5*a^5-25*a^3+20*a)*(2*a-2)!/(a+2)!^2 For example if a=, 60, then the number is, 1836520059924072508157405953466760 This agrees with the value obtained by numerical dynamical programming that is, 1836520059924072508157405953466760 Theorem Number, [3, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, a - 3], is 2 (18 a - 36 a + 66) (2 a - 3)! --------------------------------- (a - 3)! (a + 1)! (a + 3) (a + 2) and in Maple notation (18*a^2-36*a+66)*(2*a-3)!/(a-3)!/(a+1)!/(a+3)/(a+2) For example if a=, 60, then the number is, 3098219716911457914891533433362568 This agrees with the value obtained by numerical dynamical programming that is, 3098219716911457914891533433362568 Theorem Number, [3, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a - 3], is 2 2 2 14 (a - 2 a + 12) a (a + 3) (a + 4) (2 a - 2)! (a - 4) (a - 1) ----------------------------------------------------------------- 2 ((a + 4)!) and in Maple notation 14*(a^2-2*a+12)*a*(a+3)*(a+4)*(2*a-2)!/(a+4)!^2*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 4453561145315875832281709437156893 This agrees with the value obtained by numerical dynamical programming that is, 4453561145315875832281709437156893 Theorem Number, [3, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 3], is 4 3 2 40 (a - 4 a + 35 a - 62 a + 66) (2 a - 3)! ------------------------------------------------- (a - 3)! (a + 1)! (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 40*(a^4-4*a^3+35*a^2-62*a+66)*(2*a-3)!/(a-3)!/(a+1)!/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 5804732292402126309926069514785697 This agrees with the value obtained by numerical dynamical programming that is, 5804732292402126309926069514785697 Theorem Number, [3, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 4 3 2 9 (3 a - 12 a + 203 a - 382 a + 1080) a (a + 3) (a + 4) (a + 5) (a + 6) 2 2 / 2 (2 a - 2)! (a - 4) (a - 1) / ((a + 6)!) / and in Maple notation 9*(3*a^4-12*a^3+203*a^2-382*a+1080)*a*(a+3)*(a+4)*(a+5)*(a+6)*(2*a-2)!/(a+6)!^2 *(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 7070602230707679156406012920847026 This agrees with the value obtained by numerical dynamical programming that is, 7070602230707679156406012920847026 Theorem Number, [3, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 6 5 4 3 2 70 (a - 6 a + 130 a - 480 a + 2101 a - 3258 a + 2664) (2 a - 3)! --------------------------------------------------------------------- (a - 3)! (a + 1)! (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 70*(a^6-6*a^5+130*a^4-480*a^3+2101*a^2-3258*a+2664)*(2*a-3)!/(a-3)!/(a+1)!/(a+7 )/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 8193420768735632214264555472918482 This agrees with the value obtained by numerical dynamical programming that is, 8193420768735632214264555472918482 Theorem Number, [3, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 6 5 4 3 2 44 (a - 6 a + 211 a - 804 a + 6070 a - 10548 a + 20160) a (a + 3) (a + 4) 2 2 / (a + 5) (a + 6) (a + 7) (a + 8) (2 a - 2)! (a - 4) (a - 1) / / 2 ((a + 8)!) and in Maple notation 44*(a^6-6*a^5+211*a^4-804*a^3+6070*a^2-10548*a+20160)*a*(a+3)*(a+4)*(a+5)*(a+6) *(a+7)*(a+8)*(2*a-2)!/(a+8)!^2*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 9140798910196717606832700751228773 This agrees with the value obtained by numerical dynamical programming that is, 9140798910196717606832700751228773 Theorem Number, [3, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 8 7 6 5 4 3 2 108 (a - 8 a + 1022/3 a - 1932 a + 56567/3 a - 62692 a + 587528/3 a - 271368 a + 188160) (2 a - 3)!/((a - 3)! (a + 1)! (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 108*(a^8-8*a^7+1022/3*a^6-1932*a^5+56567/3*a^4-62692*a^3+587528/3*a^2-271368*a+ 188160)*(2*a-3)!/(a-3)!/(a+1)!/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 9903583243096142238530080202364273 This agrees with the value obtained by numerical dynamical programming that is, 9903583243096142238530080202364273 Theorem Number, [3, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 8 7 6 5 4 3 2 130 (a - 1) (a - 8 a + 502 a - 2900 a + 42733 a - 151748 a + 758412 a - 1220976 a + 1814400) (2 a - 3)!/((a - 3)! (a + 1)! (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 130*(a-1)*(a^8-8*a^7+502*a^6-2900*a^5+42733*a^4-151748*a^3+758412*a^2-1220976*a +1814400)*(2*a-3)!/(a-3)!/(a+1)!/(a+10)/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+ 3)/(a+2) For example if a=, 60, then the number is, 10490927179428699204937062379738608 This agrees with the value obtained by numerical dynamical programming that is, 10490927179428699204937062379738608 Theorem Number, [3, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 3], is 10 9 8 7 6 5 4 154 (a - 10 a + 735 a - 5640 a + 100443 a - 524370 a + 3435065 a 3 2 - 10348460 a + 26271756 a - 33185520 a + 20476800) (2 a - 3)!/((a - 3)! (a + 1)! (a + 11) (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 154*(a^10-10*a^9+735*a^8-5640*a^7+100443*a^6-524370*a^5+3435065*a^4-10348460*a^ 3+26271756*a^2-33185520*a+20476800)*(2*a-3)!/(a-3)!/(a+1)!/(a+11)/(a+10)/(a+9)/ (a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 10924102732985104854937602372270768 This agrees with the value obtained by numerical dynamical programming that is, 10924102732985104854937602372270768 Theorem Number, [4, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, a - 4], is (12 a - 18) (2 a - 4)! ------------------------- (a - 4)! (a + 1)! (a + 2) and in Maple notation (12*a-18)*(2*a-4)!/(a-4)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 1064559085583106436931920061162088 This agrees with the value obtained by numerical dynamical programming that is, 1064559085583106436931920061162088 Theorem Number, [4, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a - 4], is 2 2 2 2 21 (a - 3 a + 6) a (2 a - 4)! (a - 9) (a - 4) (a - 1) --------------------------------------------------------- 2 ((a + 3)!) and in Maple notation 21*(a^2-3*a+6)*a*(2*a-4)!/(a+3)!^2*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 1731804096489896796262468247645448 This agrees with the value obtained by numerical dynamical programming that is, 1731804096489896796262468247645448 Theorem Number, [4, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 4], is 2 32 (a - 3 a + 17) (a - 3/2) (2 a - 4)! ----------------------------------------- (a - 4)! (a + 1)! (a + 4) (a + 3) (a + 2) and in Maple notation 32*(a^2-3*a+17)*(a-3/2)*(2*a-4)!/(a-4)!/(a+1)!/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 2419900513987524354322096064956413 This agrees with the value obtained by numerical dynamical programming that is, 2419900513987524354322096064956413 Theorem Number, [4, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 2 2 2 45 (a - 3 a + 4) (a - 3 a + 38) a (a + 4) (a + 5) (2 a - 4)! (a - 9) 2 2 / 2 (a - 4) (a - 1) / ((a + 5)!) / and in Maple notation 45*(a^2-3*a+4)*(a^2-3*a+38)/(a+5)!^2*a*(a+4)*(a+5)*(2*a-4)!*(a^2-9)*(a^2-4)*(a^ 2-1) For example if a=, 60, then the number is, 3082975243576147273906828325274252 This agrees with the value obtained by numerical dynamical programming that is, 3082975243576147273906828325274252 Theorem Number, [4, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 4 3 2 60 (a - 3/2) (2 a - 4)! (a - 6 a + 91 a - 246 a + 664) --------------------------------------------------------- (a - 4)! (a + 1)! (a + 6) (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 60*(a-3/2)*(2*a-4)!*(a^4-6*a^3+91*a^2-246*a+664)/(a-4)!/(a+1)!/(a+6)/(a+5)/(a+4 )/(a+3)/(a+2) For example if a=, 60, then the number is, 3685770452293077200802039471017742 This agrees with the value obtained by numerical dynamical programming that is, 3685770452293077200802039471017742 Theorem Number, [4, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 6 5 4 3 2 77 (a - 9 a + 175 a - 915 a + 4144 a - 8436 a + 7920) a (a + 4) (a + 5) 2 2 2 / 2 (a + 6) (a + 7) (2 a - 4)! (a - 9) (a - 4) (a - 1) / ((a + 7)!) / and in Maple notation 77*(a^6-9*a^5+175*a^4-915*a^3+4144*a^2-8436*a+7920)*a*(a+4)*(a+5)*(a+6)*(a+7)*( 2*a-4)!/(a+7)!^2*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 4205793781604100331765370877345708 This agrees with the value obtained by numerical dynamical programming that is, 4205793781604100331765370877345708 Theorem Number, [4, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 96 (a - 3/2) (2 a - 4)! 6 5 4 3 2 (a - 9 a + 270 a - 1485 a + 10509 a - 24966 a + 45920)/((a - 4)! (a + 1)! (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 96*(a-3/2)*(2*a-4)!*(a^6-9*a^5+270*a^4-1485*a^3+10509*a^2-24966*a+45920)/(a-4)! /(a+1)!/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 4633148593754162593370184749328033 This agrees with the value obtained by numerical dynamical programming that is, 4633148593754162593370184749328033 Theorem Number, [4, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 8 7 6 5 4 3 117 (2 a - 4)! (a - 12 a + 434 a - 3528 a + 33089 a - 146748 a 2 + 490636 a - 857712 a + 685440)/((a - 4)! (a + 1)! (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 117*(2*a-4)!*(a^8-12*a^7+434*a^6-3528*a^5+33089*a^4-146748*a^3+490636*a^2-\ 857712*a+685440)/(a-4)!/(a+1)!/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 4968578114503524963462750328481208 This agrees with the value obtained by numerical dynamical programming that is, 4968578114503524963462750328481208 Theorem Number, [4, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 8 7 6 5 4 3 2 140 (a - 12 a + 618 a - 5184 a + 68601 a - 334980 a + 1620764 a - 3423408 a + 4924800) (a - 3/2) (2 a - 4)!/((a - 4)! (a + 1)! (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 140*(a^8-12*a^7+618*a^6-5184*a^5+68601*a^4-334980*a^3+1620764*a^2-3423408*a+ 4924800)*(a-3/2)*(2*a-4)!/(a-4)!/(a+1)!/(a+10)/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a +4)/(a+3)/(a+2) For example if a=, 60, then the number is, 5220492530086719559777166926702368 This agrees with the value obtained by numerical dynamical programming that is, 5220492530086719559777166926702368 Theorem Number, [4, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 4], is 10 9 8 7 6 5 165 (2 a - 4)! (a - 15 a + 900 a - 9990 a + 161301 a - 1142127 a 4 3 2 + 7520750 a - 28914780 a + 80013048 a - 124157088 a + 88300800)/( (a - 4)! (a + 1)! (a + 11) (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 165*(2*a-4)!*(a^10-15*a^9+900*a^8-9990*a^7+161301*a^6-1142127*a^5+7520750*a^4-\ 28914780*a^3+80013048*a^2-124157088*a+88300800)/(a-4)!/(a+1)!/(a+11)/(a+10)/(a+ 9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 5401753668059930613463290321013368 This agrees with the value obtained by numerical dynamical programming that is, 5401753668059930613463290321013368 Theorem Number, [5, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, a - 5], is 14 (a - 2) (2 a - 5)! ------------------------- (a - 5)! (a + 1)! (a + 2) and in Maple notation 14*(a-2)*(2*a-5)!/(a-5)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 594454646080595047403579293412448 This agrees with the value obtained by numerical dynamical programming that is, 594454646080595047403579293412448 Theorem Number, [5, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 5], is 2 2 2 12 (a - 4) (a - 4 a + 9) a (a + 2) (2 a - 4)! (a - 9) (a - 1) ---------------------------------------------------------------- 2 ((a + 3)!) and in Maple notation 12*(a-4)*(a^2-4*a+9)/(a+3)!^2*a*(a+2)*(2*a-4)!*(a^2-9)*(a^2-1) For example if a=, 60, then the number is, 939581375859969371195242148490048 This agrees with the value obtained by numerical dynamical programming that is, 939581375859969371195242148490048 Theorem Number, [5, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 2 2 2 2 2 18 (a - 4 a + 23) a (2 a - 4)! (a - 16) (a - 9) (a - 4) (a - 1) -------------------------------------------------------------------- 2 ((a + 4)!) and in Maple notation 18*(a^2-4*a+23)*a*(2*a-4)!/(a+4)!^2*(a^2-16)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 1282551063578222605463207110723413 This agrees with the value obtained by numerical dynamical programming that is, 1282551063578222605463207110723413 Theorem Number, [5, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 4 3 2 2 5 (5 a - 40 a + 355 a - 1100 a + 1524) a (a + 2) (a + 5) (2 a - 4)! (a - 9) 2 2 / 2 (a - 16) (a - 1) / ((a + 5)!) / and in Maple notation 5*(5*a^4-40*a^3+355*a^2-1100*a+1524)/(a+5)!^2*a*(a+2)*(a+5)*(2*a-4)!*(a^2-9)*(a ^2-16)*(a^2-1) For example if a=, 60, then the number is, 1602656105448592290779974408807887 This agrees with the value obtained by numerical dynamical programming that is, 1602656105448592290779974408807887 Theorem Number, [5, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 4 3 2 2 33 (a - 8 a + 119 a - 412 a + 1140) a (a + 5) (a + 6) (2 a - 4)! (a - 16) 2 2 2 / 2 (a - 9) (a - 4) (a - 1) / ((a + 6)!) / and in Maple notation 33*(a^4-8*a^3+119*a^2-412*a+1140)*a*(a+5)*(a+6)*(2*a-4)!/(a+6)!^2*(a^2-16)*(a^2 -9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 1885346272295152532358418256466903 This agrees with the value obtained by numerical dynamical programming that is, 1885346272295152532358418256466903 Theorem Number, [5, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 6 5 4 3 2 42 (a - 12 a + 230 a - 1520 a + 7509 a - 18388 a + 19860) a (a + 2) 2 2 2 / (a + 5) (a + 6) (a + 7) (2 a - 4)! (a - 9) (a - 16) (a - 1) / / 2 ((a + 7)!) and in Maple notation 42*(a^6-12*a^5+230*a^4-1520*a^3+7509*a^2-18388*a+19860)/(a+7)!^2*a*(a+2)*(a+5)* (a+6)*(a+7)*(2*a-4)!*(a^2-9)*(a^2-16)*(a^2-1) For example if a=, 60, then the number is, 2122679434759615421743305815135853 This agrees with the value obtained by numerical dynamical programming that is, 2122679434759615421743305815135853 Theorem Number, [5, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 6 5 4 3 2 52 (a - 12 a + 340 a - 2400 a + 17299 a - 50508 a + 95760) a (a + 5) 2 2 2 2 (a + 6) (a + 7) (a + 8) (2 a - 4)! (a - 16) (a - 9) (a - 4) (a - 1) / 2 / ((a + 8)!) / and in Maple notation 52*(a^6-12*a^5+340*a^4-2400*a^3+17299*a^2-50508*a+95760)*a*(a+5)*(a+6)*(a+7)*(a +8)*(2*a-4)!/(a+8)!^2*(a^2-16)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 2312701084445214793963232128449228 This agrees with the value obtained by numerical dynamical programming that is, 2312701084445214793963232128449228 Theorem Number, [5, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 8 7 6 5 4 3 2 126 (a - 16 a + 546 a - 5656 a + 55209 a - 295624 a + 1096244 a - 2250384 a + 2062080) (2 a - 5)!/((a - 5)! (a + 1)! (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 126*(a^8-16*a^7+546*a^6-5656*a^5+55209*a^4-295624*a^3+1096244*a^2-2250384*a+ 2062080)*(2*a-5)!/(a-5)!/(a+1)!/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 2458108955508977791835871394289028 This agrees with the value obtained by numerical dynamical programming that is, 2458108955508977791835871394289028 Theorem Number, [5, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 8 7 6 5 4 3 30 (a - 2) (5 a - 80 a + 3770 a - 40760 a + 533549 a - 3205352 a 2 + 16155156 a - 40662288 a + 60480000) (2 a - 5)!/((a - 5)! (a + 1)! (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 30*(a-2)*(5*a^8-80*a^7+3770*a^6-40760*a^5+533549*a^4-3205352*a^3+16155156*a^2-\ 40662288*a+60480000)*(2*a-5)!/(a-5)!/(a+1)!/(a+10)/(a+9)/(a+8)/(a+7)/(a+6)/(a+5 )/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 2564615500028409390277648726670388 This agrees with the value obtained by numerical dynamical programming that is, 2564615500028409390277648726670388 Theorem Number, [5, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 5], is 10 9 8 7 6 5 4 176 (a - 20 a + 1095 a - 15600 a + 251643 a - 2167620 a + 15275105 a 3 2 - 68893000 a + 212632956 a - 381616560 a + 309808800) (2 a - 5)!/( (a - 5)! (a + 1)! (a + 11) (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 176*(a^10-20*a^9+1095*a^8-15600*a^7+251643*a^6-2167620*a^5+15275105*a^4-\ 68893000*a^3+212632956*a^2-381616560*a+309808800)*(2*a-5)!/(a-5)!/(a+1)!/(a+11) /(a+10)/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 2639370093482188845521994788600028 This agrees with the value obtained by numerical dynamical programming that is, 2639370093482188845521994788600028 Theorem Number, [6, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, a - 6], is 7 6 5 4 3 2 (4 a - 40 a + 88 a + 176 a - 572 a - 136 a + 480 a) (2 a - 4)! -------------------------------------------------------------------- 2 ((a + 2)!) and in Maple notation (4*a^7-40*a^6+88*a^5+176*a^4-572*a^3-136*a^2+480*a)*(2*a-4)!/(a+2)!^2 For example if a=, 60, then the number is, 322118281127416035538885331405760 This agrees with the value obtained by numerical dynamical programming that is, 322118281127416035538885331405760 Theorem Number, [6, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 2 (27 a - 135 a + 342) (2 a - 6)! --------------------------------- (a - 6)! (a + 1)! (a + 3) (a + 2) and in Maple notation (27*a^2-135*a+342)*(2*a-6)!/(a-6)!/(a+1)!/(a+3)/(a+2) For example if a=, 60, then the number is, 497082527615568410121963109240752 This agrees with the value obtained by numerical dynamical programming that is, 497082527615568410121963109240752 Theorem Number, [6, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 2 2 2 2 10 (a - 5 a + 30) (a - 5) a (a + 2) (2 a - 4)! (a - 16) (a - 9) (a - 1) / 2 / ((a + 4)!) / and in Maple notation 10*(a^2-5*a+30)*(a-5)/(a+4)!^2*a*(a+2)*(2*a-4)!*(a^2-16)*(a^2-9)*(a^2-1) For example if a=, 60, then the number is, 665087968845669269806850293639125 This agrees with the value obtained by numerical dynamical programming that is, 665087968845669269806850293639125 Theorem Number, [6, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 4 3 2 2 2 2 55 (a - 10 a + 95 a - 350 a + 552) a (2 a - 6)! (a - 25) (a - 16) (a - 9) 2 2 / 2 (a - 4) (a - 1) / ((a + 5)!) / and in Maple notation 55*(a^4-10*a^3+95*a^2-350*a+552)*a*(2*a-6)!/(a+5)!^2*(a^2-25)*(a^2-16)*(a^2-9)* (a^2-4)*(a^2-1) For example if a=, 60, then the number is, 817187569485938095438730407325226 This agrees with the value obtained by numerical dynamical programming that is, 817187569485938095438730407325226 Theorem Number, [6, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 4 3 2 6 (3 a - 30 a + 455 a - 1900 a + 5532) (a + 6) a (a + 2) (2 a - 4)! 2 2 2 2 / 2 (a - 25) (a - 16) (a - 9) (a - 1) / ((a + 6)!) / and in Maple notation 6*(3*a^4-30*a^3+455*a^2-1900*a+5532)/(a+6)!^2*(a+6)*a*(a+2)*(2*a-4)!*(a^2-25)*( a^2-16)*(a^2-9)*(a^2-1) For example if a=, 60, then the number is, 947778135692229511385294141298141 This agrees with the value obtained by numerical dynamical programming that is, 947778135692229511385294141298141 Theorem Number, [6, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 2 4 3 2 91 (a - 5 a + 30) (a - 10 a + 215 a - 950 a + 1464) a (a + 6) (a + 7) 2 2 2 2 2 / 2 (2 a - 6)! (a - 25) (a - 16) (a - 9) (a - 4) (a - 1) / ((a + 7)!) / and in Maple notation 91*(a^2-5*a+30)*(a^4-10*a^3+215*a^2-950*a+1464)/(a+7)!^2*a*(a+6)*(a+7)*(2*a-6)! *(a^2-25)*(a^2-16)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 1054520731950400984823617965994176 This agrees with the value obtained by numerical dynamical programming that is, 1054520731950400984823617965994176 Theorem Number, [6, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 6 5 4 3 2 28 (a - 15 a + 421 a - 3585 a + 27154 a - 92520 a + 185184) (a + 8) a 2 2 2 2 (a + 2) (a + 6) (a + 7) (2 a - 4)! (a - 25) (a - 16) (a - 9) (a - 1) / 2 / ((a + 8)!) / and in Maple notation 28*(a^6-15*a^5+421*a^4-3585*a^3+27154*a^2-92520*a+185184)/(a+8)!^2*(a+8)*a*(a+2 )*(a+6)*(a+7)*(2*a-4)!*(a^2-25)*(a^2-16)*(a^2-9)*(a^2-1) For example if a=, 60, then the number is, 1137799785377828883605220454611516 This agrees with the value obtained by numerical dynamical programming that is, 1137799785377828883605220454611516 Theorem Number, [6, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 8 7 6 5 4 3 2 135 (a - 20 a + 2030/3 a - 8400 a + 262955/3 a - 541100 a + 6686180/3 a - 5202800 a + 5384064) (2 a - 6)!/((a - 6)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9)) and in Maple notation 135*(a^8-20*a^7+2030/3*a^6-8400*a^5+262955/3*a^4-541100*a^3+6686180/3*a^2-\ 5202800*a+5384064)*(2*a-6)!/(a-6)!/(a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/( a+8)/(a+9) For example if a=, 60, then the number is, 1199928603014163982696257231833976 This agrees with the value obtained by numerical dynamical programming that is, 1199928603014163982696257231833976 Theorem Number, [6, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 8 7 6 5 4 3 160 (a - 5/2) (a - 20 a + 910 a - 11900 a + 160405 a - 1122800 a 2 + 6034860 a - 17326800 a + 27234144) (2 a - 6)!/((a - 6)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 160*(a-5/2)*(a^8-20*a^7+910*a^6-11900*a^5+160405*a^4-1122800*a^3+6034860*a^2-\ 17326800*a+27234144)*(2*a-6)!/(a-6)!/(a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7) /(a+8)/(a+9)/(a+10) For example if a=, 60, then the number is, 1244306329897260482046997786992876 This agrees with the value obtained by numerical dynamical programming that is, 1244306329897260482046997786992876 Theorem Number, [6, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 6], is 10 9 8 7 6 5 187 (2 a - 6)! (a - 25 a + 1320 a - 22650 a + 379533 a - 3776745 a 4 3 2 + 28919330 a - 147563300 a + 505947816 a - 1022554080 a + 932601600)/( (a - 6)! (a + 1)! (a + 11) (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 187*(2*a-6)!*(a^10-25*a^9+1320*a^8-22650*a^7+379533*a^6-3776745*a^5+28919330*a^ 4-147563300*a^3+505947816*a^2-1022554080*a+932601600)/(a-6)!/(a+1)!/(a+11)/(a+2 )/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10) For example if a=, 60, then the number is, 1274683196467943437940603293763616 This agrees with the value obtained by numerical dynamical programming that is, 1274683196467943437940603293763616 Theorem Number, [7, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 18 (a - 3) (2 a - 7)! ------------------------- (a - 7)! (a + 1)! (a + 2) and in Maple notation 18*(a-3)*(2*a-7)!/(a-7)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 170162483291221949208585077242608 This agrees with the value obtained by numerical dynamical programming that is, 170162483291221949208585077242608 Theorem Number, [7, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 2 30 (a - 6 a + 17) (2 a - 7)! --------------------------------- (a - 7)! (a + 1)! (a + 3) (a + 2) and in Maple notation 30*(a^2-6*a+17)*(2*a-7)!/(a-7)!/(a+1)!/(a+3)/(a+2) For example if a=, 60, then the number is, 257226031782934135603992201141360 This agrees with the value obtained by numerical dynamical programming that is, 257226031782934135603992201141360 Theorem Number, [7, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 2 44 (a - 3) (a - 6 a + 38) (2 a - 7)! ----------------------------------------- (a - 7)! (a + 1)! (a + 4) (a + 3) (a + 2) and in Maple notation 44*(a-3)*(a^2-6*a+38)*(2*a-7)!/(a-7)!/(a+1)!/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 338167924521322808893472261640981 This agrees with the value obtained by numerical dynamical programming that is, 338167924521322808893472261640981 Theorem Number, [7, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 4 3 2 2 30 (a - 6) (a - 12 a + 123 a - 522 a + 926) a (a + 3) (2 a - 6)! (a - 16) 2 2 2 / 2 (a - 25) (a - 4) (a - 1) / ((a + 5)!) / and in Maple notation 30*(a-6)*(a^4-12*a^3+123*a^2-522*a+926)/(a+5)!^2*a*(a+3)*(2*a-6)!*(a^2-16)*(a^2 -25)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 409325632423202961235872314827461 This agrees with the value obtained by numerical dynamical programming that is, 409325632423202961235872314827461 Theorem Number, [7, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 4 3 2 2 2 39 (a - 12 a + 189 a - 918 a + 2840) a (2 a - 6)! (a - 36) (a - 25) 2 2 2 2 / 2 (a - 16) (a - 9) (a - 4) (a - 1) / ((a + 6)!) / and in Maple notation 39*(a^4-12*a^3+189*a^2-918*a+2840)*a*(2*a-6)!/(a+6)!^2*(a^2-36)*(a^2-25)*(a^2-\ 16)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 468758490727614224840035995613896 This agrees with the value obtained by numerical dynamical programming that is, 468758490727614224840035995613896 Theorem Number, [7, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 49 --- (a - 5) (a + 6) 128 6 5 4 3 2 (a - 18 a + 370 a - 3360 a + 20149 a - 64302 a + 618120/7) (a + 5) / 2 2 (2 a + 8)! (a - 6) (a - 4) (a + 7) / ((2 a + 7) ((a + 7)!) (4 a - 25) / 2 2 (4 a - 9) (4 a - 1)) and in Maple notation 49/128*(a-5)*(a+6)*(a^6-18*a^5+370*a^4-3360*a^3+20149*a^2-64302*a+618120/7)*(a+ 5)*(2*a+8)!*(a-6)*(a-4)*(a+7)/(2*a+7)/(a+7)!^2/(4*a^2-25)/(4*a^2-9)/(4*a^2-1) For example if a=, 60, then the number is, 516068228681374434674196139523496 This agrees with the value obtained by numerical dynamical programming that is, 516068228681374434674196139523496 Theorem Number, [7, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 6 5 4 3 2 60 (a - 18 a + 513 a - 5076 a + 40884 a - 157824 a + 336224) a (a + 7) 2 2 2 2 2 2 (a + 8) (2 a - 6)! (a - 36) (a - 25) (a - 16) (a - 9) (a - 4) (a - 1) / 2 / ((a + 8)!) / and in Maple notation 60*(a^6-18*a^5+513*a^4-5076*a^3+40884*a^2-157824*a+336224)*a*(a+7)*(a+8)*(2*a-6 )!/(a+8)!^2*(a^2-36)*(a^2-25)*(a^2-16)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 552037544155042123621638484231236 This agrees with the value obtained by numerical dynamical programming that is, 552037544155042123621638484231236 Theorem Number, [7, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 8 7 6 5 4 3 2 144 (a - 24 a + 826 a - 11844 a + 133189 a - 923916 a + 4200384 a - 10943496 a + 12615120) (2 a - 7)!/((a - 7)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9)) and in Maple notation 144*(a^8-24*a^7+826*a^6-11844*a^5+133189*a^4-923916*a^3+4200384*a^2-10943496*a+ 12615120)*(2*a-7)!/(a-7)!/(a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9 ) For example if a=, 60, then the number is, 578197046317709533765232916745956 This agrees with the value obtained by numerical dynamical programming that is, 578197046317709533765232916745956 Theorem Number, [7, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 8 7 6 5 4 3 170 (a - 3) (a - 24 a + 1086 a - 16524 a + 233445 a - 1846188 a 2 + 10651868 a - 34062384 a + 56972160) (2 a - 7)!/((a - 7)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 170*(a-3)*(a^8-24*a^7+1086*a^6-16524*a^5+233445*a^4-1846188*a^3+10651868*a^2-\ 34062384*a+56972160)*(2*a-7)!/(a-7)!/(a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7) /(a+8)/(a+9)/(a+10) For example if a=, 60, then the number is, 596415271038138622972379039390136 This agrees with the value obtained by numerical dynamical programming that is, 596415271038138622972379039390136 Theorem Number, [7, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 7], is 10 9 8 7 6 5 4 198 (a - 30 a + 1575 a - 31320 a + 554043 a - 6189750 a + 51553025 a 3 2 - 291807780 a + 1102051356 a - 2468326320 a + 2495203200) (2 a - 7)!/( (a - 7)! (a + 1)! (a + 11) (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 198*(a^10-30*a^9+1575*a^8-31320*a^7+554043*a^6-6189750*a^5+51553025*a^4-\ 291807780*a^3+1102051356*a^2-2468326320*a+2495203200)*(2*a-7)!/(a-7)!/(a+1)!/(a +11)/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10) For example if a=, 60, then the number is, 608573912888392489658838423516696 This agrees with the value obtained by numerical dynamical programming that is, 608573912888392489658838423516696 Theorem Number, [8, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is (20 a - 70) (2 a - 8)! ------------------------- (a - 8)! (a + 1)! (a + 2) and in Maple notation (20*a-70)*(2*a-8)!/(a-8)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 87900697996440188187670653936240 This agrees with the value obtained by numerical dynamical programming that is, 87900697996440188187670653936240 Theorem Number, [8, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 2 33 (2 a - 8)! (a - 7 a + 22) --------------------------------- (a - 8)! (a + 1)! (a + 3) (a + 2) and in Maple notation 33*(2*a-8)!*(a^2-7*a+22)/(a-8)!/(a+1)!/(a+3)/(a+2) For example if a=, 60, then the number is, 130469379891724243925248030886736 This agrees with the value obtained by numerical dynamical programming that is, 130469379891724243925248030886736 Theorem Number, [8, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 2 2 12 (a - 7 a + 47) (a - 7) (a - 6) (a - 5) (a + 3) a (2 a - 6)! (a - 16) 2 2 / 2 (a - 4) (a - 1) / ((a + 4)!) / and in Maple notation 12*(a^2-7*a+47)*(a-7)*(a-6)*(a-5)/(a+4)!^2*(a+3)*a*(2*a-6)!*(a^2-16)*(a^2-4)*(a ^2-1) For example if a=, 60, then the number is, 168842590734828861477150714435861 This agrees with the value obtained by numerical dynamical programming that is, 168842590734828861477150714435861 Theorem Number, [8, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 4 3 2 65 (2 a - 8)! (a - 14 a + 155 a - 742 a + 1464) -------------------------------------------------- (a - 8)! (a + 1)! (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 65*(2*a-8)!*(a^4-14*a^3+155*a^2-742*a+1464)/(a-8)!/(a+1)!/(a+5)/(a+4)/(a+3)/(a+ 2) For example if a=, 60, then the number is, 201627087793604396267648084073216 This agrees with the value obtained by numerical dynamical programming that is, 201627087793604396267648084073216 Theorem Number, [8, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 4 3 2 2 21 (a - 14 a + 231 a - 1274 a + 4200) (a - 7) (a + 3) a (2 a - 6)! (a - 36) 2 2 2 2 / 2 (a - 25) (a - 16) (a - 4) (a - 1) / ((a + 6)!) / and in Maple notation 21*(a^4-14*a^3+231*a^2-1274*a+4200)*(a-7)/(a+6)!^2*(a+3)*a*(2*a-6)!*(a^2-36)*(a ^2-25)*(a^2-16)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 228275449039240125081314395222296 This agrees with the value obtained by numerical dynamical programming that is, 228275449039240125081314395222296 Theorem Number, [8, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 105 --- (a - 4) (a - 5) (a - 6) (a - 7) 64 6 5 4 3 2 (a - 21 a + 455 a - 4655 a + 30576 a - 108388 a + 164784) / 2 GAMMA(2 a - 1) / ((a + 1) (a + 6) GAMMA(a) (a + 5) (a + 2) (a - 3/2) / (a + 3) (a + 4) (a - 5/2) a (a + 7) (a - 7/2)) and in Maple notation 105/64*(a-4)*(a-5)*(a-6)*(a-7)*(a^6-21*a^5+455*a^4-4655*a^3+30576*a^2-108388*a+ 164784)*GAMMA(2*a-1)/(a+1)/(a+6)/GAMMA(a)^2/(a+5)/(a+2)/(a-3/2)/(a+3)/(a+4)/(a-\ 5/2)/a/(a+7)/(a-7/2) For example if a=, 60, then the number is, 248936825747364606101808227982816 This agrees with the value obtained by numerical dynamical programming that is, 248936825747364606101808227982816 Theorem Number, [8, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 2 (a - 4) (a - 5) (a - 6) (a - 7) 6 5 4 3 2 (a - 21 a + 616 a - 6909 a + 59395 a - 254898 a + 579060) / 2 GAMMA(2 a - 1) / ((a + 1) (a + 8) (a + 6) GAMMA(a) (a + 5) (a + 2) / (a - 3/2) (a + 3) (a + 4) (a - 5/2) a (a + 7)) and in Maple notation 2*(a-4)*(a-5)*(a-6)*(a-7)*(a^6-21*a^5+616*a^4-6909*a^3+59395*a^2-254898*a+ 579060)*GAMMA(2*a-1)/(a+1)/(a+8)/(a+6)/GAMMA(a)^2/(a+5)/(a+2)/(a-3/2)/(a+3)/(a+ 4)/(a-5/2)/a/(a+7) For example if a=, 60, then the number is, 264244764512907814028756739930036 This agrees with the value obtained by numerical dynamical programming that is, 264244764512907814028756739930036 Theorem Number, [8, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 8 7 6 5 4 3 153 (2 a - 8)! (a - 28 a + 994 a - 16072 a + 194929 a - 1494892 a 2 + 7444716 a - 21370608 a + 27135360)/((a - 8)! (a + 1)! (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 153*(2*a-8)!*(a^8-28*a^7+994*a^6-16072*a^5+194929*a^4-1494892*a^3+7444716*a^2-\ 21370608*a+27135360)/(a-8)!/(a+1)!/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a +2) For example if a=, 60, then the number is, 275096327910032016245402660497536 This agrees with the value obtained by numerical dynamical programming that is, 275096327910032016245402660497536 Theorem Number, [8, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 8 7 6 5 4 3 2 180 (a - 28 a + 1282 a - 22120 a + 330001 a - 2891980 a + 17915820 a - 62859888 a + 112014720) (a - 7/2) (2 a - 8)!/((a - 8)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7) (a + 8) (a + 9) (a + 10)) and in Maple notation 180*(a^8-28*a^7+1282*a^6-22120*a^5+330001*a^4-2891980*a^3+17915820*a^2-62859888 *a+112014720)*(a-7/2)*(2*a-8)!/(a-8)!/(a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7 )/(a+8)/(a+9)/(a+10) For example if a=, 60, then the number is, 282462989233336903235902862574216 This agrees with the value obtained by numerical dynamical programming that is, 282462989233336903235902862574216 Theorem Number, [8, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 8], is 10 9 8 7 6 5 209 (2 a - 8)! (a - 35 a + 1860 a - 41790 a + 785253 a - 9676275 a 4 3 2 + 87359390 a - 541627660 a + 2234285496 a - 5483348640 a + 6078240000)/ ((a - 8)! (a + 1)! (a + 11) (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 209*(2*a-8)!*(a^10-35*a^9+1860*a^8-41790*a^7+785253*a^6-9676275*a^5+87359390*a^ 4-541627660*a^3+2234285496*a^2-5483348640*a+6078240000)/(a-8)!/(a+1)!/(a+11)/(a +10)/(a+9)/(a+8)/(a+7)/(a+6)/(a+5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 287254969760285882931862044624096 This agrees with the value obtained by numerical dynamical programming that is, 287254969760285882931862044624096 Theorem Number, [9, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 22 (a - 4) (2 a - 9)! ------------------------- (a - 9)! (a + 1)! (a + 2) and in Maple notation 22*(a-4)*(2*a-9)!/(a-9)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 44494866596428130657829746948256 This agrees with the value obtained by numerical dynamical programming that is, 44494866596428130657829746948256 Theorem Number, [9, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 2 36 (a - 8 a + 83/3) (2 a - 9)! --------------------------------- (a - 9)! (a + 1)! (a + 3) (a + 2) and in Maple notation 36*(a^2-8*a+83/3)*(2*a-9)!/(a-9)!/(a+1)!/(a+3)/(a+2) For example if a=, 60, then the number is, 64960579046083926685511178174456 This agrees with the value obtained by numerical dynamical programming that is, 64960579046083926685511178174456 Theorem Number, [9, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 2 52 (a - 4) (a - 8 a + 57) (2 a - 9)! ----------------------------------------- (a - 9)! (a + 1)! (a + 4) (a + 3) (a + 2) and in Maple notation 52*(a-4)*(a^2-8*a+57)*(2*a-9)!/(a-9)!/(a+1)!/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 82868077439532748209732430497381 This agrees with the value obtained by numerical dynamical programming that is, 82868077439532748209732430497381 Theorem Number, [9, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 4 3 2 70 (a - 16 a + 191 a - 1016 a + 2208) (2 a - 9)! --------------------------------------------------- (a - 9)! (a + 1)! (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 70*(a^4-16*a^3+191*a^2-1016*a+2208)*(2*a-9)!/(a-9)!/(a+1)!/(a+5)/(a+4)/(a+3)/(a +2) For example if a=, 60, then the number is, 97745076104859461476008547811811 This agrees with the value obtained by numerical dynamical programming that is, 97745076104859461476008547811811 Theorem Number, [9, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 4 3 2 90 (a - 16 a + 833/3 a - 5128/3 a + 6004) (a - 4) (2 a - 9)! --------------------------------------------------------------- (a - 9)! (a + 1)! (a + 6) (a + 5) (a + 4) (a + 3) (a + 2) and in Maple notation 90*(a^4-16*a^3+833/3*a^2-5128/3*a+6004)*(a-4)*(2*a-9)!/(a-9)!/(a+1)!/(a+6)/(a+5 )/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 109516438685168477023398741646461 This agrees with the value obtained by numerical dynamical programming that is, 109516438685168477023398741646461 Theorem Number, [9, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 6 5 4 3 2 56 (a - 8) (a - 24 a + 550 a - 6240 a + 44629 a - 173736 a + 289620) a 2 2 2 2 2 2 (a + 4) (2 a - 8)! (a - 25) (a - 36) (a - 49) (a - 9) (a - 4) (a - 1) / 2 / ((a + 7)!) / and in Maple notation 56*(a-8)*(a^6-24*a^5+550*a^4-6240*a^3+44629*a^2-173736*a+289620)/(a+7)!^2*a*(a+ 4)*(2*a-8)!*(a^2-25)*(a^2-36)*(a^2-49)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 118406452812983942496502380572331 This agrees with the value obtained by numerical dynamical programming that is, 118406452812983942496502380572331 Theorem Number, [9, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 17 -- (a - 4) (a - 5) (a - 6) (a - 7) (a - 8) 16 6 5 4 3 2 (a - 24 a + 730 a - 9120 a + 83689 a - 394056 a + 953820) / 2 GAMMA(2 a - 1) / ((a + 1) (a + 8) (a + 6) GAMMA(a) (a - 3/2) (a + 5) / (a + 2) (a + 3) (a - 5/2) (a + 4) a (a - 7/2) (a + 7)) and in Maple notation 17/16*(a-4)*(a-5)*(a-6)*(a-7)*(a-8)*(a^6-24*a^5+730*a^4-9120*a^3+83689*a^2-\ 394056*a+953820)*GAMMA(2*a-1)/(a+1)/(a+8)/(a+6)/GAMMA(a)^2/(a-3/2)/(a+5)/(a+2)/ (a+3)/(a-5/2)/(a+4)/a/(a-7/2)/(a+7) For example if a=, 60, then the number is, 124824377450711684950347253593681 This agrees with the value obtained by numerical dynamical programming that is, 124824377450711684950347253593681 Theorem Number, [9, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 81 8 7 6 5 -- (a - 7) (a - 5) GAMMA(2 a - 1) (a - 6) (a - 32 a + 3542/3 a - 21168 a 64 4 3 2 + 828947/3 a - 2316048 a + 37632788/3 a - 39293152 a + 54456640) / 2 (a - 8) / ((a + 1) (a + 8) (a + 6) GAMMA(a) (a - 3/2) (a + 5) (a + 2) / (a + 9) (a + 3) (a - 5/2) (a + 4) a (a - 7/2) (a + 7)) and in Maple notation 81/64*(a-7)*(a-5)*GAMMA(2*a-1)*(a-6)*(a^8-32*a^7+3542/3*a^6-21168*a^5+828947/3* a^4-2316048*a^3+37632788/3*a^2-39293152*a+54456640)*(a-8)/(a+1)/(a+8)/(a+6)/ GAMMA(a)^2/(a-3/2)/(a+5)/(a+2)/(a+9)/(a+3)/(a-5/2)/(a+4)/a/(a-7/2)/(a+7) For example if a=, 60, then the number is, 129258016210108144713148301139831 This agrees with the value obtained by numerical dynamical programming that is, 129258016210108144713148301139831 Theorem Number, [9, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 8 7 6 5 4 3 190 (a - 4) (a - 32 a + 1498 a - 28784 a + 454657 a - 4357136 a 2 + 28918980 a - 110175264 a + 208928160) (2 a - 9)!/((a - 9)! (a + 1)! (a + 10) (a + 9) (a + 8) (a + 7) (a + 6) (a + 5) (a + 4) (a + 3) (a + 2)) and in Maple notation 190*(a-4)*(a^8-32*a^7+1498*a^6-28784*a^5+454657*a^4-4357136*a^3+28918980*a^2-\ 110175264*a+208928160)*(2*a-9)!/(a-9)!/(a+1)!/(a+10)/(a+9)/(a+8)/(a+7)/(a+6)/(a +5)/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 132191038774016571940847455670361 This agrees with the value obtained by numerical dynamical programming that is, 132191038774016571940847455670361 Theorem Number, [9, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 9], is 10 9 8 7 110 (a - 5) (a - 6) (a - 7) (a - 8) (a - 40 a + 2175 a - 54240 a 6 5 4 3 2 + 1084251 a - 14560392 a + 141832025 a - 954427280 a + 4268795148 a / - 11380127328 a + 13713779520) (2 a - 8)! / (a (a + 4) (a + 5) (a + 6) / 2 2 2 (a + 7) (a + 8) (a + 9) (a + 10) (a + 11) ((a - 4)!) (a - 9) (a - 4) 2 (a - 1)) and in Maple notation 110*(a-5)*(a-6)*(a-7)*(a-8)*(a^10-40*a^9+2175*a^8-54240*a^7+1084251*a^6-\ 14560392*a^5+141832025*a^4-954427280*a^3+4268795148*a^2-11380127328*a+ 13713779520)*(2*a-8)!/a/(a+4)/(a+5)/(a+6)/(a+7)/(a+8)/(a+9)/(a+10)/(a+11)/(a-4) !^2/(a^2-9)/(a^2-4)/(a^2-1) For example if a=, 60, then the number is, 134049996737057124409107483189711 This agrees with the value obtained by numerical dynamical programming that is, 134049996737057124409107483189711 Theorem Number, [10, 1], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is (24 a - 108) (2 a - 10)! -------------------------- (a - 10)! (a + 1)! (a + 2) and in Maple notation (24*a-108)*(2*a-10)!/(a-10)!/(a+1)!/(a+2) For example if a=, 60, then the number is, 22102969445628259709895945724296 This agrees with the value obtained by numerical dynamical programming that is, 22102969445628259709895945724296 Theorem Number, [10, 2], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 2 39 (2 a - 10)! (a - 9 a + 34) ---------------------------------- (a - 10)! (a + 1)! (a + 2) (a + 3) and in Maple notation 39*(2*a-10)!*(a^2-9*a+34)/(a-10)!/(a+1)!/(a+2)/(a+3) For example if a=, 60, then the number is, 31782698306951947020285811844796 This agrees with the value obtained by numerical dynamical programming that is, 31782698306951947020285811844796 Theorem Number, [10, 3], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 2 56 (a - 9 a + 68) (a - 9/2) (2 a - 10)! ------------------------------------------ (a + 1)! (a - 10)! (a + 4) (a + 3) (a + 2) and in Maple notation 56*(a^2-9*a+68)*(a-9/2)*(2*a-10)!/(a+1)!/(a-10)!/(a+4)/(a+3)/(a+2) For example if a=, 60, then the number is, 40010467839077081234117198047221 This agrees with the value obtained by numerical dynamical programming that is, 40010467839077081234117198047221 Theorem Number, [10, 4], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 4 3 2 75 (a - 18 a + 231 a - 1350 a + 16024/5) (2 a - 10)! ------------------------------------------------------- (a - 10)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) and in Maple notation 75*(a^4-18*a^3+231*a^2-1350*a+16024/5)*(2*a-10)!/(a-10)!/(a+1)!/(a+2)/(a+3)/(a+ 4)/(a+5) For example if a=, 60, then the number is, 46659696972278660286561929159226 This agrees with the value obtained by numerical dynamical programming that is, 46659696972278660286561929159226 Theorem Number, [10, 5], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 4 3 2 24 (a - 18 a + 329 a - 2232 a + 8340) (a - 9) (a - 8) (a - 7) (a + 4) a 2 2 2 2 2 / 2 (2 a - 8)! (a - 36) (a - 25) (a - 9) (a - 4) (a - 1) / ((a + 6)!) / and in Maple notation 24*(a^4-18*a^3+329*a^2-2232*a+8340)*(a-9)*(a-8)*(a-7)/(a+6)!^2*(a+4)*a*(2*a-8)! *(a^2-36)*(a^2-25)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 51781830419386096781507391881871 This agrees with the value obtained by numerical dynamical programming that is, 51781830419386096781507391881871 Theorem Number, [10, 6], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 6 5 4 3 2 119 (2 a - 10)! (a - 27 a + 655 a - 8145 a + 63064 a - 267228 a + 484560)/ ((a - 10)! (a + 1)! (a + 2) (a + 3) (a + 4) (a + 5) (a + 6) (a + 7)) and in Maple notation 119*(2*a-10)!*(a^6-27*a^5+655*a^4-8145*a^3+63064*a^2-267228*a+484560)/(a-10)!/( a+1)!/(a+2)/(a+3)/(a+4)/(a+5)/(a+6)/(a+7) For example if a=, 60, then the number is, 55549711100094125759665568085096 This agrees with the value obtained by numerical dynamical programming that is, 55549711100094125759665568085096 Theorem Number, [10, 7], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 6 5 4 3 2 36 (a - 27 a + 855 a - 11745 a + 114864 a - 587628 a + 1512560) (a - 9) 2 2 2 2 2 (a + 4) a (2 a - 8)! (a - 64) (a - 49) (a - 36) (a - 25) (a - 9) 2 2 / 2 (a - 4) (a - 1) / ((a + 8)!) / and in Maple notation 36*(a^6-27*a^5+855*a^4-11745*a^3+114864*a^2-587628*a+1512560)*(a-9)/(a+8)!^2*(a +4)*a*(2*a-8)!*(a^2-64)*(a^2-49)*(a^2-36)*(a^2-25)*(a^2-9)*(a^2-4)*(a^2-1) For example if a=, 60, then the number is, 58199755057113839235352264903221 This agrees with the value obtained by numerical dynamical programming that is, 58199755057113839235352264903221 Theorem Number, [10, 8], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 171 8 7 6 5 --- (a - 5) (a - 6) (a - 7) (a - 8) (a - 9) (a - 36 a + 1386 a - 27216 a 256 4 3 2 + 381129 a - 3461724 a + 20261324 a - 68724144 a + 103178880) / 2 GAMMA(2 a - 1) / ((a + 1) (a + 8) (a + 6) (a - 7/2) GAMMA(a) (a + 5) / (a + 2) (a - 5/2) (a + 9) (a + 3) (a - 9/2) (a + 4) (a - 3/2) a (a + 7)) and in Maple notation 171/256*(a-5)*(a-6)*(a-7)*(a-8)*(a-9)*(a^8-36*a^7+1386*a^6-27216*a^5+381129*a^4 -3461724*a^3+20261324*a^2-68724144*a+103178880)*GAMMA(2*a-1)/(a+1)/(a+8)/(a+6)/ (a-7/2)/GAMMA(a)^2/(a+5)/(a+2)/(a-5/2)/(a+9)/(a+3)/(a-9/2)/(a+4)/(a-3/2)/a/(a+7 ) For example if a=, 60, then the number is, 59983349859490585522466615631246 This agrees with the value obtained by numerical dynamical programming that is, 59983349859490585522466615631246 Theorem Number, [10, 9], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 25 8 7 6 5 4 -- (a - 9) (a - 7) (a - 5) (a - 36 a + 1734 a - 36612 a + 3062049/5 a 32 3 2 - 31781592/5 a + 225303196/5 a - 924807168/5 a + 372464064) / GAMMA(2 a - 1) (a - 6) (a - 8) / ((a + 1) (a + 8) (a + 6) (a - 7/2) / 2 GAMMA(a) (a + 10) (a + 5) (a + 2) (a - 5/2) (a + 9) (a + 3) (a + 4) (a - 3/2) a (a + 7)) and in Maple notation 25/32*(a-9)*(a-7)*(a-5)*(a^8-36*a^7+1734*a^6-36612*a^5+3062049/5*a^4-31781592/5 *a^3+225303196/5*a^2-924807168/5*a+372464064)*GAMMA(2*a-1)*(a-6)*(a-8)/(a+1)/(a +8)/(a+6)/(a-7/2)/GAMMA(a)^2/(a+10)/(a+5)/(a+2)/(a-5/2)/(a+9)/(a+3)/(a+4)/(a-3/ 2)/a/(a+7) For example if a=, 60, then the number is, 61132777621022266463051419433751 This agrees with the value obtained by numerical dynamical programming that is, 61132777621022266463051419433751 Theorem Number, [10, 10], The number of ways of playing Stanley Solitaire starting with position, [a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a - 10], is 10 9 8 7 6 5 4 231 (a - 45 a + 2520 a - 68850 a + 1463133 a - 21225645 a + 222023330 a 3 2 - 1610140500 a + 7758834216 a - 22311905760 a + 29023660800) (2 a - 10)! / 2 2 2 2 2 2 2 / ((a - 4) (a - 1) (a - 49) (a - 9) (a - 36) ((a - 10)!) (a - 25) / 2 2 2 (a - 81) (a - 64) (a + 11) (a - 16) (a + 10) a) and in Maple notation 231*(a^10-45*a^9+2520*a^8-68850*a^7+1463133*a^6-21225645*a^5+222023330*a^4-\ 1610140500*a^3+7758834216*a^2-22311905760*a+29023660800)/(a^2-4)/(a^2-1)/(a^2-\ 49)/(a^2-9)/(a^2-36)*(2*a-10)!/(a-10)!^2/(a^2-25)/(a^2-81)/(a^2-64)/(a+11)/(a^2 -16)/(a+10)/a For example if a=, 60, then the number is, 61842307822531137990726643150596 This agrees with the value obtained by numerical dynamical programming that is, 61842307822531137990726643150596 --------------------------- This concludes this paper that took, 9.118, seconds to produce