n ----- \ Pi (2 j - 1) (2 i) evaluations of , ) sin(------------) , for i from 1 to , 50 / 4 n ----- j = 1 By Shalosh B. Ekhad Comment: It can be easily shown that for each positive integer i n - 1 ----- \ Pi (2 j - 1) (2 i) binomial(2 i - 1, i) n ) sin(------------) = ---------------------- / 4 n (2 i - 1) ----- 2 j = 1 This paper confirms it directly for i from 1 to , 50 ------------------------------ n ----- \ Pi (2 j - 1) 2 Theorem , 1, : for n>=, 1, we have :, ) sin(------------) = n/2 / 4 n ----- j = 1 and in Maple notation: 1/2*n ------------------------------ n ----- \ Pi (2 j - 1) 4 3 n Theorem , 2, : for n>=, 2, we have :, ) sin(------------) = --- / 4 n 8 ----- j = 1 and in Maple notation: 3/8*n ------------------------------ n ----- \ Pi (2 j - 1) 6 5 n Theorem , 3, : for n>=, 2, we have :, ) sin(------------) = --- / 4 n 16 ----- j = 1 and in Maple notation: 5/16*n ------------------------------ n ----- \ Pi (2 j - 1) 8 35 n Theorem , 4, : for n>=, 3, we have :, ) sin(------------) = ---- / 4 n 128 ----- j = 1 and in Maple notation: 35/128*n ------------------------------ n ----- \ Pi (2 j - 1) 10 63 n Theorem , 5, : for n>=, 3, we have :, ) sin(------------) = ---- / 4 n 256 ----- j = 1 and in Maple notation: 63/256*n ------------------------------ n ----- \ Pi (2 j - 1) 12 231 n Theorem , 6, : for n>=, 4, we have :, ) sin(------------) = ----- / 4 n 1024 ----- j = 1 and in Maple notation: 231/1024*n ------------------------------ n ----- \ Pi (2 j - 1) 14 429 n Theorem , 7, : for n>=, 4, we have :, ) sin(------------) = ----- / 4 n 2048 ----- j = 1 and in Maple notation: 429/2048*n ------------------------------ n ----- \ Pi (2 j - 1) 16 6435 n Theorem , 8, : for n>=, 5, we have :, ) sin(------------) = ------ / 4 n 32768 ----- j = 1 and in Maple notation: 6435/32768*n ------------------------------ n ----- \ Pi (2 j - 1) 18 12155 n Theorem , 9, : for n>=, 5, we have :, ) sin(------------) = ------- / 4 n 65536 ----- j = 1 and in Maple notation: 12155/65536*n ------------------------------ n ----- \ Pi (2 j - 1) 20 46189 n Theorem , 10, : for n>=, 6, we have :, ) sin(------------) = ------- / 4 n 262144 ----- j = 1 and in Maple notation: 46189/262144*n ------------------------------ n ----- \ Pi (2 j - 1) 22 88179 n Theorem , 11, : for n>=, 6, we have :, ) sin(------------) = ------- / 4 n 524288 ----- j = 1 and in Maple notation: 88179/524288*n ------------------------------ n ----- \ Pi (2 j - 1) 24 676039 n Theorem , 12, : for n>=, 7, we have :, ) sin(------------) = -------- / 4 n 4194304 ----- j = 1 and in Maple notation: 676039/4194304*n ------------------------------ n ----- \ Pi (2 j - 1) 26 1300075 n Theorem , 13, : for n>=, 7, we have :, ) sin(------------) = --------- / 4 n 8388608 ----- j = 1 and in Maple notation: 1300075/8388608*n ------------------------------ n ----- \ Pi (2 j - 1) 28 5014575 n Theorem , 14, : for n>=, 8, we have :, ) sin(------------) = --------- / 4 n 33554432 ----- j = 1 and in Maple notation: 5014575/33554432*n ------------------------------ n ----- \ Pi (2 j - 1) 30 9694845 n Theorem , 15, : for n>=, 8, we have :, ) sin(------------) = --------- / 4 n 67108864 ----- j = 1 and in Maple notation: 9694845/67108864*n ------------------------------ n ----- \ Pi (2 j - 1) 32 300540195 n Theorem , 16, : for n>=, 9, we have :, ) sin(------------) = ----------- / 4 n 2147483648 ----- j = 1 and in Maple notation: 300540195/2147483648*n ------------------------------ n ----- \ Pi (2 j - 1) 34 583401555 n Theorem , 17, : for n>=, 9, we have :, ) sin(------------) = ----------- / 4 n 4294967296 ----- j = 1 and in Maple notation: 583401555/4294967296*n ------------------------------ Theorem , 18, : for n>=, 10, we have :, n ----- \ Pi (2 j - 1) 36 2268783825 n ) sin(------------) = ------------ / 4 n 17179869184 ----- j = 1 and in Maple notation: 2268783825/17179869184*n ------------------------------ Theorem , 19, : for n>=, 10, we have :, n ----- \ Pi (2 j - 1) 38 4418157975 n ) sin(------------) = ------------ / 4 n 34359738368 ----- j = 1 and in Maple notation: 4418157975/34359738368*n ------------------------------ Theorem , 20, : for n>=, 11, we have :, n ----- \ Pi (2 j - 1) 40 34461632205 n ) sin(------------) = ------------- / 4 n 274877906944 ----- j = 1 and in Maple notation: 34461632205/274877906944*n ------------------------------ Theorem , 21, : for n>=, 11, we have :, n ----- \ Pi (2 j - 1) 42 67282234305 n ) sin(------------) = ------------- / 4 n 549755813888 ----- j = 1 and in Maple notation: 67282234305/549755813888*n ------------------------------ Theorem , 22, : for n>=, 12, we have :, n ----- \ Pi (2 j - 1) 44 263012370465 n ) sin(------------) = -------------- / 4 n 2199023255552 ----- j = 1 and in Maple notation: 263012370465/2199023255552*n ------------------------------ Theorem , 23, : for n>=, 12, we have :, n ----- \ Pi (2 j - 1) 46 514589420475 n ) sin(------------) = -------------- / 4 n 4398046511104 ----- j = 1 and in Maple notation: 514589420475/4398046511104*n ------------------------------ Theorem , 24, : for n>=, 13, we have :, n ----- \ Pi (2 j - 1) 48 8061900920775 n ) sin(------------) = --------------- / 4 n 70368744177664 ----- j = 1 and in Maple notation: 8061900920775/70368744177664*n ------------------------------ Theorem , 25, : for n>=, 13, we have :, n ----- \ Pi (2 j - 1) 50 15801325804719 n ) sin(------------) = ---------------- / 4 n 140737488355328 ----- j = 1 and in Maple notation: 15801325804719/140737488355328*n ------------------------------ Theorem , 26, : for n>=, 14, we have :, n ----- \ Pi (2 j - 1) 52 61989816618513 n ) sin(------------) = ---------------- / 4 n 562949953421312 ----- j = 1 and in Maple notation: 61989816618513/562949953421312*n ------------------------------ Theorem , 27, : for n>=, 14, we have :, n ----- \ Pi (2 j - 1) 54 121683714103007 n ) sin(------------) = ----------------- / 4 n 1125899906842624 ----- j = 1 and in Maple notation: 121683714103007/1125899906842624*n ------------------------------ Theorem , 28, : for n>=, 15, we have :, n ----- \ Pi (2 j - 1) 56 956086325095055 n ) sin(------------) = ----------------- / 4 n 9007199254740992 ----- j = 1 and in Maple notation: 956086325095055/9007199254740992*n ------------------------------ Theorem , 29, : for n>=, 15, we have :, n ----- \ Pi (2 j - 1) 58 1879204156221315 n ) sin(------------) = ------------------ / 4 n 18014398509481984 ----- j = 1 and in Maple notation: 1879204156221315/18014398509481984*n ------------------------------ Theorem , 30, : for n>=, 16, we have :, n ----- \ Pi (2 j - 1) 60 7391536347803839 n ) sin(------------) = ------------------ / 4 n 72057594037927936 ----- j = 1 and in Maple notation: 7391536347803839/72057594037927936*n ------------------------------ Theorem , 31, : for n>=, 16, we have :, n ----- \ Pi (2 j - 1) 62 14544636039226909 n ) sin(------------) = ------------------- / 4 n 144115188075855872 ----- j = 1 and in Maple notation: 14544636039226909/144115188075855872*n ------------------------------ Theorem , 32, : for n>=, 17, we have :, n ----- \ Pi (2 j - 1) 64 916312070471295267 n ) sin(------------) = -------------------- / 4 n 9223372036854775808 ----- j = 1 and in Maple notation: 916312070471295267/9223372036854775808*n ------------------------------ Theorem , 33, : for n>=, 17, we have :, n ----- \ Pi (2 j - 1) 66 1804857108504066435 n ) sin(------------) = --------------------- / 4 n 18446744073709551616 ----- j = 1 and in Maple notation: 1804857108504066435/18446744073709551616*n ------------------------------ Theorem , 34, : for n>=, 18, we have :, n ----- \ Pi (2 j - 1) 68 7113260368810144185 n ) sin(------------) = --------------------- / 4 n 73786976294838206464 ----- j = 1 and in Maple notation: 7113260368810144185/73786976294838206464*n ------------------------------ Theorem , 35, : for n>=, 18, we have :, n ----- \ Pi (2 j - 1) 70 14023284727082855679 n ) sin(------------) = ---------------------- / 4 n 147573952589676412928 ----- j = 1 and in Maple notation: 14023284727082855679/147573952589676412928*n ------------------------------ Theorem , 36, : for n>=, 19, we have :, n ----- \ Pi (2 j - 1) 72 110628135069209194801 n ) sin(------------) = ----------------------- / 4 n 1180591620717411303424 ----- j = 1 and in Maple notation: 110628135069209194801/1180591620717411303424*n ------------------------------ Theorem , 37, : for n>=, 19, we have :, n ----- \ Pi (2 j - 1) 74 218266320541953276229 n ) sin(------------) = ----------------------- / 4 n 2361183241434822606848 ----- j = 1 and in Maple notation: 218266320541953276229/2361183241434822606848*n ------------------------------ Theorem , 38, : for n>=, 20, we have :, n ----- \ Pi (2 j - 1) 76 861577581086657669325 n ) sin(------------) = ----------------------- / 4 n 9444732965739290427392 ----- j = 1 and in Maple notation: 861577581086657669325/9444732965739290427392*n ------------------------------ Theorem , 39, : for n>=, 20, we have :, n ----- \ Pi (2 j - 1) 78 1701063429324939500975 n ) sin(------------) = ------------------------ / 4 n 18889465931478580854784 ----- j = 1 and in Maple notation: 1701063429324939500975/18889465931478580854784*n ------------------------------ Theorem , 40, : for n>=, 21, we have :, n ----- \ Pi (2 j - 1) 80 26876802183334044115405 n ) sin(------------) = ------------------------- / 4 n 302231454903657293676544 ----- j = 1 and in Maple notation: 26876802183334044115405/302231454903657293676544*n ------------------------------ Theorem , 41, : for n>=, 21, we have :, n ----- \ Pi (2 j - 1) 82 53098072606098965203605 n ) sin(------------) = ------------------------- / 4 n 604462909807314587353088 ----- j = 1 and in Maple notation: 53098072606098965203605/604462909807314587353088*n ------------------------------ Theorem , 42, : for n>=, 22, we have :, n ----- \ Pi (2 j - 1) 84 209863810776486386280915 n ) sin(------------) = -------------------------- / 4 n 2417851639229258349412352 ----- j = 1 and in Maple notation: 209863810776486386280915/2417851639229258349412352*n ------------------------------ Theorem , 43, : for n>=, 22, we have :, n ----- \ Pi (2 j - 1) 86 414847067813984717066925 n ) sin(------------) = -------------------------- / 4 n 4835703278458516698824704 ----- j = 1 and in Maple notation: 414847067813984717066925/4835703278458516698824704*n ------------------------------ Theorem , 44, : for n>=, 23, we have :, n ----- \ Pi (2 j - 1) 88 3281063172710606398620225 n ) sin(------------) = --------------------------- / 4 n 38685626227668133590597632 ----- j = 1 and in Maple notation: 3281063172710606398620225/38685626227668133590597632*n ------------------------------ Theorem , 45, : for n>=, 23, we have :, n ----- \ Pi (2 j - 1) 90 6489213830472088210604445 n ) sin(------------) = --------------------------- / 4 n 77371252455336267181195264 ----- j = 1 and in Maple notation: 6489213830472088210604445/77371252455336267181195264*n ------------------------------ Theorem , 46, : for n>=, 24, we have :, n ----- \ Pi (2 j - 1) 92 25674715590128696833261065 n ) sin(------------) = ---------------------------- / 4 n 309485009821345068724781056 ----- j = 1 and in Maple notation: 25674715590128696833261065/309485009821345068724781056*n ------------------------------ Theorem , 47, : for n>=, 24, we have :, n ----- \ Pi (2 j - 1) 94 50803160635786570329644235 n ) sin(------------) = ---------------------------- / 4 n 618970019642690137449562112 ----- j = 1 and in Maple notation: 50803160635786570329644235/618970019642690137449562112*n ------------------------------ Theorem , 48, : for n>=, 25, we have :, n ----- \ Pi (2 j - 1) 96 1608766753466574727105400775 n ) sin(------------) = ------------------------------ / 4 n 19807040628566084398385987584 ----- j = 1 and in Maple notation: 1608766753466574727105400775/19807040628566084398385987584*n ------------------------------ Theorem , 49, : for n>=, 25, we have :, n ----- \ Pi (2 j - 1) 98 3184701532372607112841303575 n ) sin(------------) = ------------------------------ / 4 n 39614081257132168796771975168 ----- j = 1 and in Maple notation: 3184701532372607112841303575/39614081257132168796771975168*n ------------------------------ Theorem , 50, : for n>=, 26, we have :, n ----- \ Pi (2 j - 1) 100 12611418068195524166851562157 n ) sin(------------) = ------------------------------- / 4 n 158456325028528675187087900672 ----- j = 1 and in Maple notation: 12611418068195524166851562157/158456325028528675187087900672*n ------------------------------ This ends this book that took, 0.320, seconds to generate