The Number of Ways of Having n Coins in Your Two pockets, with denominatio\ ns in the set, {1, 5, 10, 25, 50, 100}, in such a way That both pockets carry the same amount By Shalosh B. Ekhad Theorem: Let a(n) be the number of ways of having n Coins in your two\ pockets, with denominations in the set, {1, 5, 10, 25, 50, 100}, in such a way that both pockets carry the same amount , then infinity ----- \ n 228 226 225 224 223 222 ) a(n) t = - (t + 4 t + 5 t + 14 t + 28 t + 56 t / ----- n = 0 221 220 219 218 217 216 + 106 t + 191 t + 328 t + 538 t + 863 t + 1327 t 215 214 213 212 211 210 + 1986 t + 2906 t + 4141 t + 5785 t + 7924 t + 10658 t 209 208 207 206 205 + 14093 t + 18377 t + 23603 t + 29923 t + 37493 t 204 203 202 201 200 + 46440 t + 56917 t + 69091 t + 83091 t + 99069 t 199 198 197 196 195 + 117206 t + 137589 t + 160371 t + 185706 t + 213668 t 194 193 192 191 190 + 244352 t + 277891 t + 314304 t + 353639 t + 395976 t 189 188 187 186 185 + 441275 t + 489528 t + 540755 t + 594850 t + 651750 t 184 183 182 181 180 + 711434 t + 773743 t + 838535 t + 905788 t + 975299 t 179 178 177 176 175 + 1046913 t + 1120545 t + 1196000 t + 1273120 t + 1351815 t 174 173 172 171 170 + 1431880 t + 1513125 t + 1595529 t + 1678865 t + 1762962 t 169 168 167 166 165 + 1847777 t + 1933141 t + 2018879 t + 2104955 t + 2191205 t 164 163 162 161 160 + 2277475 t + 2363733 t + 2449809 t + 2535535 t + 2620905 t 159 158 157 156 155 + 2705730 t + 2789825 t + 2873145 t + 2955551 t + 3036814 t 154 153 152 151 150 + 3116879 t + 3195558 t + 3272672 t + 3348145 t + 3421783 t 149 148 147 146 145 + 3493383 t + 3562884 t + 3630149 t + 3694951 t + 3757252 t 144 143 142 141 140 + 3816928 t + 3873832 t + 3927933 t + 3979154 t + 4027407 t 139 138 137 136 135 + 4072708 t + 4115047 t + 4154374 t + 4190791 t + 4224340 t 134 133 132 131 130 + 4255020 t + 4282970 t + 4308303 t + 4331105 t + 4351488 t 129 128 127 126 125 + 4369609 t + 4385581 t + 4399596 t + 4411783 t + 4422238 t 124 123 122 121 120 + 4431174 t + 4438743 t + 4445065 t + 4450255 t + 4454477 t 119 118 117 116 115 + 4457831 t + 4460438 t + 4462365 t + 4463676 t + 4464449 t 114 113 112 111 110 + 4464714 t + 4464449 t + 4463676 t + 4462365 t + 4460438 t 109 108 107 106 105 + 4457831 t + 4454477 t + 4450255 t + 4445065 t + 4438743 t 104 103 102 101 100 + 4431174 t + 4422238 t + 4411783 t + 4399596 t + 4385581 t 99 98 97 96 95 + 4369609 t + 4351488 t + 4331105 t + 4308303 t + 4282970 t 94 93 92 91 90 + 4255020 t + 4224340 t + 4190791 t + 4154374 t + 4115047 t 89 88 87 86 85 + 4072708 t + 4027407 t + 3979154 t + 3927933 t + 3873832 t 84 83 82 81 80 + 3816928 t + 3757252 t + 3694951 t + 3630149 t + 3562884 t 79 78 77 76 75 + 3493383 t + 3421783 t + 3348145 t + 3272672 t + 3195558 t 74 73 72 71 70 + 3116879 t + 3036814 t + 2955551 t + 2873145 t + 2789825 t 69 68 67 66 65 + 2705730 t + 2620905 t + 2535535 t + 2449809 t + 2363733 t 64 63 62 61 60 + 2277475 t + 2191205 t + 2104955 t + 2018879 t + 1933141 t 59 58 57 56 55 + 1847777 t + 1762962 t + 1678865 t + 1595529 t + 1513125 t 54 53 52 51 50 + 1431880 t + 1351815 t + 1273120 t + 1196000 t + 1120545 t 49 48 47 46 45 + 1046913 t + 975299 t + 905788 t + 838535 t + 773743 t 44 43 42 41 40 + 711434 t + 651750 t + 594850 t + 540755 t + 489528 t 39 38 37 36 35 + 441275 t + 395976 t + 353639 t + 314304 t + 277891 t 34 33 32 31 30 + 244352 t + 213668 t + 185706 t + 160371 t + 137589 t 29 28 27 26 25 24 + 117206 t + 99069 t + 83091 t + 69091 t + 56917 t + 46440 t 23 22 21 20 19 18 + 37493 t + 29923 t + 23603 t + 18377 t + 14093 t + 10658 t 17 16 15 14 13 12 + 7924 t + 5785 t + 4141 t + 2906 t + 1986 t + 1327 t 11 10 9 8 7 6 5 4 + 863 t + 538 t + 328 t + 191 t + 106 t + 56 t + 28 t + 14 t 3 2 / 11 100 99 98 97 96 95 + 5 t + 4 t + 1) / ((t - 1) (t + t + t + t + t + t / 94 93 92 91 90 89 88 87 86 85 84 83 + t + t + t + t + t + t + t + t + t + t + t + t 82 81 80 79 78 77 76 75 74 73 72 71 + t + t + t + t + t + t + t + t + t + t + t + t 70 69 68 67 66 65 64 63 62 61 60 59 + t + t + t + t + t + t + t + t + t + t + t + t 58 57 56 55 54 53 52 51 50 49 48 47 + t + t + t + t + t + t + t + t + t + t + t + t 46 45 44 43 42 41 40 39 38 37 36 35 + t + t + t + t + t + t + t + t + t + t + t + t 34 33 32 31 30 29 28 27 26 25 24 23 + t + t + t + t + t + t + t + t + t + t + t + t 22 21 20 19 18 17 16 15 14 13 12 11 + t + t + t + t + t + t + t + t + t + t + t + t 10 9 8 7 6 5 4 3 2 16 15 14 + t + t + t + t + t + t + t + t + t + t + 1) (t + t + t 13 12 11 10 9 8 7 6 5 4 3 2 + t + t + t + t + t + t + t + t + t + t + t + t + t + 1) 2 5 32 31 29 28 26 25 23 22 20 19 (t + t + 1) (t - t + t - t + t - t + t - t + t - t 17 16 15 13 12 10 9 7 6 4 3 + t - t + t - t + t - t + t - t + t - t + t - t + 1) 12 11 10 9 8 7 6 5 4 3 2 5 (t + t + t + t + t + t + t + t + t + t + t + t + 1) (1 + t) 12 11 10 9 8 7 6 5 4 3 2 (t - t + t - t + t - t + t - t + t - t + t - t + 1) 10 9 8 7 6 5 4 3 2 2 2 2 (t + t + t + t + t + t + t + t + t + t + 1) (t - t + 1) 6 5 4 3 2 (t + t + t + t + t + t + 1) 12 11 9 8 6 4 3 (t - t + t - t + t - t + t - t + 1)) and in Maple notation -(t^228+4*t^226+5*t^225+14*t^224+28*t^223+56*t^222+106*t^221+191*t^220+328*t^ 219+538*t^218+863*t^217+1327*t^216+1986*t^215+2906*t^214+4141*t^213+5785*t^212+ 7924*t^211+10658*t^210+14093*t^209+18377*t^208+23603*t^207+29923*t^206+37493*t^ 205+46440*t^204+56917*t^203+69091*t^202+83091*t^201+99069*t^200+117206*t^199+ 137589*t^198+160371*t^197+185706*t^196+213668*t^195+244352*t^194+277891*t^193+ 314304*t^192+353639*t^191+395976*t^190+441275*t^189+489528*t^188+540755*t^187+ 594850*t^186+651750*t^185+711434*t^184+773743*t^183+838535*t^182+905788*t^181+ 975299*t^180+1046913*t^179+1120545*t^178+1196000*t^177+1273120*t^176+1351815*t^ 175+1431880*t^174+1513125*t^173+1595529*t^172+1678865*t^171+1762962*t^170+ 1847777*t^169+1933141*t^168+2018879*t^167+2104955*t^166+2191205*t^165+2277475*t ^164+2363733*t^163+2449809*t^162+2535535*t^161+2620905*t^160+2705730*t^159+ 2789825*t^158+2873145*t^157+2955551*t^156+3036814*t^155+3116879*t^154+3195558*t ^153+3272672*t^152+3348145*t^151+3421783*t^150+3493383*t^149+3562884*t^148+ 3630149*t^147+3694951*t^146+3757252*t^145+3816928*t^144+3873832*t^143+3927933*t ^142+3979154*t^141+4027407*t^140+4072708*t^139+4115047*t^138+4154374*t^137+ 4190791*t^136+4224340*t^135+4255020*t^134+4282970*t^133+4308303*t^132+4331105*t ^131+4351488*t^130+4369609*t^129+4385581*t^128+4399596*t^127+4411783*t^126+ 4422238*t^125+4431174*t^124+4438743*t^123+4445065*t^122+4450255*t^121+4454477*t ^120+4457831*t^119+4460438*t^118+4462365*t^117+4463676*t^116+4464449*t^115+ 4464714*t^114+4464449*t^113+4463676*t^112+4462365*t^111+4460438*t^110+4457831*t ^109+4454477*t^108+4450255*t^107+4445065*t^106+4438743*t^105+4431174*t^104+ 4422238*t^103+4411783*t^102+4399596*t^101+4385581*t^100+4369609*t^99+4351488*t^ 98+4331105*t^97+4308303*t^96+4282970*t^95+4255020*t^94+4224340*t^93+4190791*t^ 92+4154374*t^91+4115047*t^90+4072708*t^89+4027407*t^88+3979154*t^87+3927933*t^ 86+3873832*t^85+3816928*t^84+3757252*t^83+3694951*t^82+3630149*t^81+3562884*t^ 80+3493383*t^79+3421783*t^78+3348145*t^77+3272672*t^76+3195558*t^75+3116879*t^ 74+3036814*t^73+2955551*t^72+2873145*t^71+2789825*t^70+2705730*t^69+2620905*t^ 68+2535535*t^67+2449809*t^66+2363733*t^65+2277475*t^64+2191205*t^63+2104955*t^ 62+2018879*t^61+1933141*t^60+1847777*t^59+1762962*t^58+1678865*t^57+1595529*t^ 56+1513125*t^55+1431880*t^54+1351815*t^53+1273120*t^52+1196000*t^51+1120545*t^ 50+1046913*t^49+975299*t^48+905788*t^47+838535*t^46+773743*t^45+711434*t^44+ 651750*t^43+594850*t^42+540755*t^41+489528*t^40+441275*t^39+395976*t^38+353639* t^37+314304*t^36+277891*t^35+244352*t^34+213668*t^33+185706*t^32+160371*t^31+ 137589*t^30+117206*t^29+99069*t^28+83091*t^27+69091*t^26+56917*t^25+46440*t^24+ 37493*t^23+29923*t^22+23603*t^21+18377*t^20+14093*t^19+10658*t^18+7924*t^17+ 5785*t^16+4141*t^15+2906*t^14+1986*t^13+1327*t^12+863*t^11+538*t^10+328*t^9+191 *t^8+106*t^7+56*t^6+28*t^5+14*t^4+5*t^3+4*t^2+1)/(t-1)^11/(t^100+t^99+t^98+t^97 +t^96+t^95+t^94+t^93+t^92+t^91+t^90+t^89+t^88+t^87+t^86+t^85+t^84+t^83+t^82+t^ 81+t^80+t^79+t^78+t^77+t^76+t^75+t^74+t^73+t^72+t^71+t^70+t^69+t^68+t^67+t^66+t ^65+t^64+t^63+t^62+t^61+t^60+t^59+t^58+t^57+t^56+t^55+t^54+t^53+t^52+t^51+t^50+ t^49+t^48+t^47+t^46+t^45+t^44+t^43+t^42+t^41+t^40+t^39+t^38+t^37+t^36+t^35+t^34 +t^33+t^32+t^31+t^30+t^29+t^28+t^27+t^26+t^25+t^24+t^23+t^22+t^21+t^20+t^19+t^ 18+t^17+t^16+t^15+t^14+t^13+t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1) /(t^16+t^15+t^14+t^13+t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+ t+1)^5/(t^32-t^31+t^29-t^28+t^26-t^25+t^23-t^22+t^20-t^19+t^17-t^16+t^15-t^13+t ^12-t^10+t^9-t^7+t^6-t^4+t^3-t+1)/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t ^2+t+1)/(1+t)^5/(t^12-t^11+t^10-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1)/(t^10+t^9+ t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)^2/(t^2-t+1)^2/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^12- t^11+t^9-t^8+t^6-t^4+t^3-t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 1, 0, 6, 6, 25, 44, 108, 202, 422, 762, 1422, 2474, 4266, 7078, 11572, 18366, 28729, 43966, 66266, 98198, 143569, 206832, 294508, 414162, 576228, 793242, 1081686, 1461108, 1957138, 2599758, 3427128 Furthermore, a(n) is a quasi-polynomial given as sum of, 12, quasi-polynomials 12 ----- \ a(n) = ) P[i](n) / ----- i = 1 where , P[1](n), P[2](n), P[3](n), P[4](n), P[5](n), P[6](n), P[7](n), P[8](n), P[9](n), P[10](n), P[11](n), P[12](n), are defined as followed P[1](n), is the polynomial 225668543 10 225668543 9 1967373871 8 ------------------ n + ---------------- n + ---------------- n 266738894514892800 4445648241914880 1317229108715520 458284985 7 1102141471331 6 50945450827 5 + -------------- n + ---------------- n + -------------- n 16465363858944 3175463029939200 17641461277440 1824373471710809 4 77790296942725 3 1769619648721019 2 + ------------------ n + --------------- n + ---------------- n 106695557805957120 889129648382976 5388664535654400 14966259050129 101657511171107 + -------------- n - ---------------- 47043896739840 2124255449530368 and in Maple notation 225668543/266738894514892800*n^10+225668543/4445648241914880*n^9+1967373871/ 1317229108715520*n^8+458284985/16465363858944*n^7+1102141471331/ 3175463029939200*n^6+50945450827/17641461277440*n^5+1824373471710809/ 106695557805957120*n^4+77790296942725/889129648382976*n^3+1769619648721019/ 5388664535654400*n^2+14966259050129/47043896739840*n-101657511171107/ 2124255449530368 This is the leading term in particular, a(n) , is asymptotic to 10 225668543 n ------------------ 266738894514892800 P[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 31 4 31 3 11161 2 9673 554550353 [- ------- n - ------ n - ------ n - ----- n - ---------, 2875392 119808 958464 79872 5750784 31 4 31 3 11161 2 9673 554550353 ------- n + ------ n + ------ n + ----- n + ---------] 2875392 119808 958464 79872 5750784 and in Maple format [-31/2875392*n^4-31/119808*n^3-11161/958464*n^2-9673/79872*n-554550353/5750784, 31/2875392*n^4+31/119808*n^3+11161/958464*n^2+9673/79872*n+554550353/5750784] P[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 461 4 205 3 37 2 355553 9164651 [-------- n + ------- n - ----- n + -------- n + --------, 37476432 6246072 30618 37476432 74952864 461 4 43 3 2144 2 1212607 2456147 -------- n + ----- n + ------ n + -------- n + --------, 37476432 77112 260253 37476432 74952864 461 4 461 3 3659 2 1210 830057 - -------- n - ------ n - ------ n - ----- n - -------] 18738216 780759 520506 28917 5353776 and in Maple format [461/37476432*n^4+205/6246072*n^3-37/30618*n^2+355553/37476432*n+9164651/ 74952864, 461/37476432*n^4+43/77112*n^3+2144/260253*n^2+1212607/37476432*n+ 2456147/74952864, -461/18738216*n^4-461/780759*n^3-3659/520506*n^2-1210/28917*n -830057/5353776] P[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 n 11 n 13 -1 n 11 n 13 [- --- - ---, - --- - ---, ---, --- + ---, --- + ---, 1/432] 432 864 432 864 432 432 864 432 864 and in Maple format [-1/432*n-11/864, -1/432*n-13/864, -1/432, 1/432*n+11/864, 1/432*n+13/864, 1/ 432] P[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -20 -2 -2 -20 [8/49, ---, 2/21, --, --, 2/21, ---] 147 49 49 147 and in Maple format [8/49, -20/147, 2/21, -2/49, -2/49, 2/21, -20/147] P[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 11 12 n 1282 10 n 1402 8 n 1744 4 n 1798 1568 4 n 1270 [---- + ----, - ---- - ----, --- + ----, - --- - ----, ----, --- - ----, 121 1331 121 1331 121 1331 121 1331 1331 121 1331 8 n 688 10 n 82 12 n 302 14 n 710 14 n 1138 - --- + ----, ---- - ----, - ---- - ----, ---- + ----, - ---- - ----] 121 1331 121 1331 121 1331 121 1331 121 1331 and in Maple format [12/121*n+1282/1331, -10/121*n-1402/1331, 8/121*n+1744/1331, -4/121*n-1798/1331 , 1568/1331, 4/121*n-1270/1331, -8/121*n+688/1331, 10/121*n-82/1331, -12/121*n-\ 302/1331, 14/121*n+710/1331, -14/121*n-1138/1331] P[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 13 -1 -1 -2 -1 -1 [0, 1/13, --, 1/13, --, 1/13, --, 1/13, --, 1/13, --, 1/13, 0] 13 13 13 13 13 and in Maple format [0, 1/13, -1/13, 1/13, -1/13, 1/13, -2/13, 1/13, -1/13, 1/13, -1/13, 1/13, 0] P[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 17 -2 -4 -4 -2 [2/51, 2/51, 2/51, 2/51, 0, 0, --, 2/51, --, 4/51, 0, 4/51, --, 2/51, --, 0, 0] 17 51 51 17 and in Maple format [2/51, 2/51, 2/51, 2/51, 0, 0, -2/17, 2/51, -4/51, 4/51, 0, 4/51, -4/51, 2/51, -2/17, 0, 0] P[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 21 26 -32 -32 26 -68 100 -82 [-18/7, --, ---, -2/7, -2/7, ---, --, -18/7, 8/3, ---, 4, -26/7, ---, ---, 36/7, 21 21 21 21 21 21 21 -82 100 -68 ---, ---, -26/7, 4, ---, 8/3] 21 21 21 and in Maple format [-18/7, 26/21, -32/21, -2/7, -2/7, -32/21, 26/21, -18/7, 8/3, -68/21, 4, -26/7, 100/21, -82/21, 36/7, -82/21, 100/21, -26/7, 4, -68/21, 8/3] P[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 26 17 11 11 22 11 11 17 -8 -5 -3 [8/13, 5/13, 3/13, --, --, --, --, --, --, --, 3/13, 5/13, 8/13, --, --, --, 13 13 13 13 13 13 13 13 13 13 -17 -11 -11 -22 -11 -11 -17 -3 -5 -8 ---, ---, ---, ---, ---, ---, ---, --, --, --] 13 13 13 13 13 13 13 13 13 13 and in Maple format [8/13, 5/13, 3/13, 17/13, 11/13, 11/13, 22/13, 11/13, 11/13, 17/13, 3/13, 5/13, 8/13, -8/13, -5/13, -3/13, -17/13, -11/13, -11/13, -22/13, -11/13, -11/13, -17/ 13, -3/13, -5/13, -8/13] P[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 51 -494 -728 -536 -722 -166 -118 -392 -116 -76 302 556 688 [----, ----, ----, ----, ----, -12, ----, ----, ----, ---, 4, ---, ---, ---, 51 51 51 51 17 17 51 51 51 51 51 51 298 336 382 1216 1264 1264 1216 382 336 298 688 556 302 -76 ---, ---, ---, ----, ----, ----, ----, ---, ---, ---, ---, ---, ---, 4, ---, 17 17 17 51 51 51 51 17 17 17 51 51 51 51 -116 -392 -118 -166 -722 -536 -728 -494 -216 -132 -180 -296 ----, ----, ----, -12, ----, ----, ----, ----, ----, ----, ----, ----, ----, 51 51 17 17 51 51 51 51 17 17 17 51 -440 -226 -226 -440 -296 -180 -132 -216 ----, ----, -8, ----, ----, ----, ----, ----, ----] 51 51 51 51 51 17 17 17 and in Maple format [-494/51, -728/51, -536/51, -722/51, -166/17, -12, -118/17, -392/51, -116/51, -\ 76/51, 4, 302/51, 556/51, 688/51, 298/17, 336/17, 382/17, 1216/51, 1264/51, 1264/51, 1216/51, 382/17, 336/17, 298/17, 688/51, 556/51, 302/51, 4, -76/51, -\ 116/51, -392/51, -118/17, -12, -166/17, -722/51, -536/51, -728/51, -494/51, -\ 216/17, -132/17, -180/17, -296/51, -440/51, -226/51, -8, -226/51, -440/51, -296 /51, -180/17, -132/17, -216/17] P[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 101 10718 -8074 10592 -7994 10132 -8062 9258 -8486 8302 -9414 7192 [-----, -----, -----, -----, -----, -----, ----, -----, ----, -----, ----, 101 101 101 101 101 101 101 101 101 101 101 -10250 5736 -10970 4510 -11684 3654 -11746 2826 -11026 2570 ------, ----, ------, ----, ------, ----, ------, ----, ------, ----, 101 101 101 101 101 101 101 101 101 101 -10008 2978 -8444 3386 -6414 3996 -4612 4924 -3018 5472 -1470 ------, ----, -----, ----, -----, ----, -----, ----, -----, ----, -----, 101 101 101 101 101 101 101 101 101 101 101 5688 -420 5818 196 5534 798 4736 1318 3984 1658 3180 2276 2276 ----, ----, ----, ---, ----, ---, ----, ----, ----, ----, ----, ----, ----, 101 101 101 101 101 101 101 101 101 101 101 101 101 3180 1658 3984 1318 4736 798 5534 196 5818 -420 5688 -1470 5472 ----, ----, ----, ----, ----, ---, ----, ---, ----, ----, ----, -----, ----, 101 101 101 101 101 101 101 101 101 101 101 101 101 -3018 4924 -4612 3996 -6414 3386 -8444 2978 -10008 2570 -11026 -----, ----, -----, ----, -----, ----, -----, ----, ------, ----, ------, 101 101 101 101 101 101 101 101 101 101 101 2826 -11746 3654 -11684 4510 -10970 5736 -10250 7192 -9414 8302 ----, ------, ----, ------, ----, ------, ----, ------, ----, -----, ----, 101 101 101 101 101 101 101 101 101 101 101 -8486 9258 -8062 10132 -7994 10592 -8074 10718 -8442 10828 -9028 -----, ----, -----, -----, -----, -----, -----, -----, -----, -----, -----, 101 101 101 101 101 101 101 101 101 101 101 10790 -9386 10634 -9472 10634 -9386 10790 -9028 10828 -8442 -----, -----, -----, -----, -----, -----, -----, -----, -----, -----] 101 101 101 101 101 101 101 101 101 101 and in Maple format [10718/101, -8074/101, 10592/101, -7994/101, 10132/101, -8062/101, 9258/101, -\ 8486/101, 8302/101, -9414/101, 7192/101, -10250/101, 5736/101, -10970/101, 4510 /101, -11684/101, 3654/101, -11746/101, 2826/101, -11026/101, 2570/101, -10008/ 101, 2978/101, -8444/101, 3386/101, -6414/101, 3996/101, -4612/101, 4924/101, -\ 3018/101, 5472/101, -1470/101, 5688/101, -420/101, 5818/101, 196/101, 5534/101, 798/101, 4736/101, 1318/101, 3984/101, 1658/101, 3180/101, 2276/101, 2276/101, 3180/101, 1658/101, 3984/101, 1318/101, 4736/101, 798/101, 5534/101, 196/101, 5818/101, -420/101, 5688/101, -1470/101, 5472/101, -3018/101, 4924/101, -4612/ 101, 3996/101, -6414/101, 3386/101, -8444/101, 2978/101, -10008/101, 2570/101, -11026/101, 2826/101, -11746/101, 3654/101, -11684/101, 4510/101, -10970/101, 5736/101, -10250/101, 7192/101, -9414/101, 8302/101, -8486/101, 9258/101, -8062 /101, 10132/101, -7994/101, 10592/101, -8074/101, 10718/101, -8442/101, 10828/ 101, -9028/101, 10790/101, -9386/101, 10634/101, -9472/101, 10634/101, -9386/ 101, 10790/101, -9028/101, 10828/101, -8442/101] That makes it very easy to compute a(n) for large n. In particular, the numb\ er of ways of having, 100000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000, coins in your two pocekts so that both pockets have the same amount is 8460278858484969686400027587481741134475429587687986471116158708841620664289599\ 9247341405107342506652768988352096668115530090948506324642963101912223300137586\ 9496066721723512944703743107164733692999544864446723491398318876916928650725331\ 5657363863823683993894811013456169807057515394694169112746221269224249093833209\ 6262600382625415526049712302656509827512227057088691296515025251629589519546170\ 9278477147757907857467974284718725009833693271805731251696597271511646443278002\ 5331084566067095340126730127724153979928224226811155739550187045564241754656521\ 8309257533182000532771298197254632508077605900266491157885904788751521189055844\ 0671487042002296660867611852117917622279034662621812705220266290148605719568485\ 5203582260435678944870874542107045583557774666915667124205459105449778860931574\ 9504162035311339057980969811949569329809895623920744660850624969186425913071707\ 6439026992833281232637332060500776451892260640891495262335396457741154084426755\ 9204848140713404799726285905404906629648029 ------------------------------------ This ends this article, that took, 160.809, seconds to generate.