n ----- \ Pi j (2 i) evaluations of , ) sin(-------) , for i from 1 to , 50 / 2 n + 1 ----- j = 1 By Shalosh B. Ekhad Comment: It can be easily shown that for each positive integer i n ----- \ Pi j (2 i) binomial(2 i, i) (n + 1/2) ) sin(-------) = -------------------------- / 2 n + 1 i ----- 4 j = 1 This paper confirms it, via a different method for i from 1 to, 50 ------------------------------ n ----- \ Pi j 2 Theorem , 1, : for n>=, 1, we have :, ) sin(-------) = n/2 + 1/4 / 2 n + 1 ----- j = 1 and in Maple notation: 1/2*n+1/4 ------------------------------ n ----- \ Pi j 4 3 n Theorem , 2, : for n>=, 1, we have :, ) sin(-------) = --- + 3/16 / 2 n + 1 8 ----- j = 1 and in Maple notation: 3/8*n+3/16 ------------------------------ n ----- \ Pi j 6 5 n Theorem , 3, : for n>=, 2, we have :, ) sin(-------) = --- + 5/32 / 2 n + 1 16 ----- j = 1 and in Maple notation: 5/16*n+5/32 ------------------------------ n ----- \ Pi j 8 35 n 35 Theorem , 4, : for n>=, 2, we have :, ) sin(-------) = ---- + --- / 2 n + 1 128 256 ----- j = 1 and in Maple notation: 35/128*n+35/256 ------------------------------ n ----- \ Pi j 10 63 n 63 Theorem , 5, : for n>=, 3, we have :, ) sin(-------) = ---- + --- / 2 n + 1 256 512 ----- j = 1 and in Maple notation: 63/256*n+63/512 ------------------------------ n ----- \ Pi j 12 231 n 231 Theorem , 6, : for n>=, 3, we have :, ) sin(-------) = ----- + ---- / 2 n + 1 1024 2048 ----- j = 1 and in Maple notation: 231/1024*n+231/2048 ------------------------------ n ----- \ Pi j 14 429 429 n Theorem , 7, : for n>=, 4, we have :, ) sin(-------) = ---- + ----- / 2 n + 1 4096 2048 ----- j = 1 and in Maple notation: 429/4096+429/2048*n ------------------------------ n ----- \ Pi j 16 6435 6435 n Theorem , 8, : for n>=, 4, we have :, ) sin(-------) = ----- + ------ / 2 n + 1 65536 32768 ----- j = 1 and in Maple notation: 6435/65536+6435/32768*n ------------------------------ n ----- \ Pi j 18 12155 12155 n Theorem , 9, : for n>=, 5, we have :, ) sin(-------) = ------ + ------- / 2 n + 1 131072 65536 ----- j = 1 and in Maple notation: 12155/131072+12155/65536*n ------------------------------ n ----- \ Pi j 20 46189 46189 n Theorem , 10, : for n>=, 5, we have :, ) sin(-------) = ------ + ------- / 2 n + 1 524288 262144 ----- j = 1 and in Maple notation: 46189/524288+46189/262144*n ------------------------------ Theorem , 11, : for n>=, 6, we have :, n ----- \ Pi j 22 88179 88179 n ) sin(-------) = ------- + ------- / 2 n + 1 1048576 524288 ----- j = 1 and in Maple notation: 88179/1048576+88179/524288*n ------------------------------ Theorem , 12, : for n>=, 6, we have :, n ----- \ Pi j 24 676039 676039 n ) sin(-------) = ------- + -------- / 2 n + 1 8388608 4194304 ----- j = 1 and in Maple notation: 676039/8388608+676039/4194304*n ------------------------------ Theorem , 13, : for n>=, 7, we have :, n ----- \ Pi j 26 1300075 1300075 n ) sin(-------) = -------- + --------- / 2 n + 1 16777216 8388608 ----- j = 1 and in Maple notation: 1300075/16777216+1300075/8388608*n ------------------------------ Theorem , 14, : for n>=, 7, we have :, n ----- \ Pi j 28 5014575 n 5014575 ) sin(-------) = --------- + -------- / 2 n + 1 33554432 67108864 ----- j = 1 and in Maple notation: 5014575/33554432*n+5014575/67108864 ------------------------------ Theorem , 15, : for n>=, 8, we have :, n ----- \ Pi j 30 9694845 9694845 n ) sin(-------) = --------- + --------- / 2 n + 1 134217728 67108864 ----- j = 1 and in Maple notation: 9694845/134217728+9694845/67108864*n ------------------------------ Theorem , 16, : for n>=, 8, we have :, n ----- \ Pi j 32 300540195 300540195 n ) sin(-------) = ---------- + ----------- / 2 n + 1 4294967296 2147483648 ----- j = 1 and in Maple notation: 300540195/4294967296+300540195/2147483648*n ------------------------------ Theorem , 17, : for n>=, 9, we have :, n ----- \ Pi j 34 583401555 583401555 n ) sin(-------) = ---------- + ----------- / 2 n + 1 8589934592 4294967296 ----- j = 1 and in Maple notation: 583401555/8589934592+583401555/4294967296*n ------------------------------ Theorem , 18, : for n>=, 9, we have :, n ----- \ Pi j 36 2268783825 2268783825 n ) sin(-------) = ----------- + ------------ / 2 n + 1 34359738368 17179869184 ----- j = 1 and in Maple notation: 2268783825/34359738368+2268783825/17179869184*n ------------------------------ Theorem , 19, : for n>=, 10, we have :, n ----- \ Pi j 38 4418157975 4418157975 n ) sin(-------) = ----------- + ------------ / 2 n + 1 68719476736 34359738368 ----- j = 1 and in Maple notation: 4418157975/68719476736+4418157975/34359738368*n ------------------------------ Theorem , 20, : for n>=, 10, we have :, n ----- \ Pi j 40 34461632205 34461632205 n ) sin(-------) = ------------ + ------------- / 2 n + 1 549755813888 274877906944 ----- j = 1 and in Maple notation: 34461632205/549755813888+34461632205/274877906944*n ------------------------------ Theorem , 21, : for n>=, 11, we have :, n ----- \ Pi j 42 67282234305 67282234305 n ) sin(-------) = ------------- + ------------- / 2 n + 1 1099511627776 549755813888 ----- j = 1 and in Maple notation: 67282234305/1099511627776+67282234305/549755813888*n ------------------------------ Theorem , 22, : for n>=, 11, we have :, n ----- \ Pi j 44 263012370465 263012370465 n ) sin(-------) = ------------- + -------------- / 2 n + 1 4398046511104 2199023255552 ----- j = 1 and in Maple notation: 263012370465/4398046511104+263012370465/2199023255552*n ------------------------------ Theorem , 23, : for n>=, 12, we have :, n ----- \ Pi j 46 514589420475 514589420475 n ) sin(-------) = ------------- + -------------- / 2 n + 1 8796093022208 4398046511104 ----- j = 1 and in Maple notation: 514589420475/8796093022208+514589420475/4398046511104*n ------------------------------ Theorem , 24, : for n>=, 12, we have :, n ----- \ Pi j 48 8061900920775 8061900920775 n ) sin(-------) = --------------- + --------------- / 2 n + 1 140737488355328 70368744177664 ----- j = 1 and in Maple notation: 8061900920775/140737488355328+8061900920775/70368744177664*n ------------------------------ Theorem , 25, : for n>=, 13, we have :, n ----- \ Pi j 50 15801325804719 15801325804719 n ) sin(-------) = --------------- + ---------------- / 2 n + 1 281474976710656 140737488355328 ----- j = 1 and in Maple notation: 15801325804719/281474976710656+15801325804719/140737488355328*n ------------------------------ Theorem , 26, : for n>=, 13, we have :, n ----- \ Pi j 52 61989816618513 61989816618513 n ) sin(-------) = ---------------- + ---------------- / 2 n + 1 1125899906842624 562949953421312 ----- j = 1 and in Maple notation: 61989816618513/1125899906842624+61989816618513/562949953421312*n ------------------------------ Theorem , 27, : for n>=, 14, we have :, n ----- \ Pi j 54 121683714103007 121683714103007 n ) sin(-------) = ---------------- + ----------------- / 2 n + 1 2251799813685248 1125899906842624 ----- j = 1 and in Maple notation: 121683714103007/2251799813685248+121683714103007/1125899906842624*n ------------------------------ Theorem , 28, : for n>=, 14, we have :, n ----- \ Pi j 56 956086325095055 956086325095055 n ) sin(-------) = ----------------- + ----------------- / 2 n + 1 18014398509481984 9007199254740992 ----- j = 1 and in Maple notation: 956086325095055/18014398509481984+956086325095055/9007199254740992*n ------------------------------ Theorem , 29, : for n>=, 15, we have :, n ----- \ Pi j 58 1879204156221315 1879204156221315 n ) sin(-------) = ----------------- + ------------------ / 2 n + 1 36028797018963968 18014398509481984 ----- j = 1 and in Maple notation: 1879204156221315/36028797018963968+1879204156221315/18014398509481984*n ------------------------------ Theorem , 30, : for n>=, 15, we have :, n ----- \ Pi j 60 7391536347803839 7391536347803839 n ) sin(-------) = ------------------ + ------------------ / 2 n + 1 144115188075855872 72057594037927936 ----- j = 1 and in Maple notation: 7391536347803839/144115188075855872+7391536347803839/72057594037927936*n ------------------------------ Theorem , 31, : for n>=, 16, we have :, n ----- \ Pi j 62 14544636039226909 14544636039226909 n ) sin(-------) = ------------------ + ------------------- / 2 n + 1 288230376151711744 144115188075855872 ----- j = 1 and in Maple notation: 14544636039226909/288230376151711744+14544636039226909/144115188075855872*n ------------------------------ Theorem , 32, : for n>=, 16, we have :, n ----- \ Pi j 64 916312070471295267 916312070471295267 n ) sin(-------) = -------------------- + -------------------- / 2 n + 1 18446744073709551616 9223372036854775808 ----- j = 1 and in Maple notation: 916312070471295267/18446744073709551616+916312070471295267/9223372036854775808* n ------------------------------ Theorem , 33, : for n>=, 17, we have :, n ----- \ Pi j 66 1804857108504066435 1804857108504066435 n ) sin(-------) = -------------------- + --------------------- / 2 n + 1 36893488147419103232 18446744073709551616 ----- j = 1 and in Maple notation: 1804857108504066435/36893488147419103232+1804857108504066435/ 18446744073709551616*n ------------------------------ Theorem , 34, : for n>=, 17, we have :, n ----- \ Pi j 68 7113260368810144185 7113260368810144185 n ) sin(-------) = --------------------- + --------------------- / 2 n + 1 147573952589676412928 73786976294838206464 ----- j = 1 and in Maple notation: 7113260368810144185/147573952589676412928+7113260368810144185/ 73786976294838206464*n ------------------------------ Theorem , 35, : for n>=, 18, we have :, n ----- \ Pi j 70 14023284727082855679 14023284727082855679 n ) sin(-------) = --------------------- + ---------------------- / 2 n + 1 295147905179352825856 147573952589676412928 ----- j = 1 and in Maple notation: 14023284727082855679/295147905179352825856+14023284727082855679/ 147573952589676412928*n ------------------------------ Theorem , 36, : for n>=, 18, we have :, n ----- \ Pi j 72 110628135069209194801 110628135069209194801 n ) sin(-------) = ---------------------- + ----------------------- / 2 n + 1 2361183241434822606848 1180591620717411303424 ----- j = 1 and in Maple notation: 110628135069209194801/2361183241434822606848+110628135069209194801/ 1180591620717411303424*n ------------------------------ Theorem , 37, : for n>=, 19, we have :, n ----- \ Pi j 74 218266320541953276229 218266320541953276229 n ) sin(-------) = ---------------------- + ----------------------- / 2 n + 1 4722366482869645213696 2361183241434822606848 ----- j = 1 and in Maple notation: 218266320541953276229/4722366482869645213696+218266320541953276229/ 2361183241434822606848*n ------------------------------ Theorem , 38, : for n>=, 19, we have :, n ----- \ Pi j 76 861577581086657669325 861577581086657669325 n ) sin(-------) = ----------------------- + ----------------------- / 2 n + 1 18889465931478580854784 9444732965739290427392 ----- j = 1 and in Maple notation: 861577581086657669325/18889465931478580854784+861577581086657669325/ 9444732965739290427392*n ------------------------------ Theorem , 39, : for n>=, 20, we have :, n ----- \ Pi j 78 1701063429324939500975 1701063429324939500975 n ) sin(-------) = ----------------------- + ------------------------ / 2 n + 1 37778931862957161709568 18889465931478580854784 ----- j = 1 and in Maple notation: 1701063429324939500975/37778931862957161709568+1701063429324939500975/ 18889465931478580854784*n ------------------------------ Theorem , 40, : for n>=, 20, we have :, n ----- \ Pi j 80 26876802183334044115405 26876802183334044115405 n ) sin(-------) = ------------------------ + ------------------------- / 2 n + 1 604462909807314587353088 302231454903657293676544 ----- j = 1 and in Maple notation: 26876802183334044115405/604462909807314587353088+26876802183334044115405/ 302231454903657293676544*n ------------------------------ n ----- \ Pi j 82 Theorem , 41, : for n>=, 21, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 53098072606098965203605 53098072606098965203605 n ------------------------- + ------------------------- 1208925819614629174706176 604462909807314587353088 and in Maple notation: 53098072606098965203605/1208925819614629174706176+53098072606098965203605/ 604462909807314587353088*n ------------------------------ n ----- \ Pi j 84 Theorem , 42, : for n>=, 21, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 209863810776486386280915 209863810776486386280915 n ------------------------- + -------------------------- 4835703278458516698824704 2417851639229258349412352 and in Maple notation: 209863810776486386280915/4835703278458516698824704+209863810776486386280915/ 2417851639229258349412352*n ------------------------------ n ----- \ Pi j 86 Theorem , 43, : for n>=, 22, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 414847067813984717066925 414847067813984717066925 n ------------------------- + -------------------------- 9671406556917033397649408 4835703278458516698824704 and in Maple notation: 414847067813984717066925/9671406556917033397649408+414847067813984717066925/ 4835703278458516698824704*n ------------------------------ n ----- \ Pi j 88 Theorem , 44, : for n>=, 22, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 3281063172710606398620225 3281063172710606398620225 n -------------------------- + --------------------------- 77371252455336267181195264 38685626227668133590597632 and in Maple notation: 3281063172710606398620225/77371252455336267181195264+3281063172710606398620225/ 38685626227668133590597632*n ------------------------------ n ----- \ Pi j 90 Theorem , 45, : for n>=, 23, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 6489213830472088210604445 6489213830472088210604445 n --------------------------- + --------------------------- 154742504910672534362390528 77371252455336267181195264 and in Maple notation: 6489213830472088210604445/154742504910672534362390528+6489213830472088210604445 /77371252455336267181195264*n ------------------------------ n ----- \ Pi j 92 Theorem , 46, : for n>=, 23, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 25674715590128696833261065 25674715590128696833261065 n --------------------------- + ---------------------------- 618970019642690137449562112 309485009821345068724781056 and in Maple notation: 25674715590128696833261065/618970019642690137449562112+ 25674715590128696833261065/309485009821345068724781056*n ------------------------------ n ----- \ Pi j 94 Theorem , 47, : for n>=, 24, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 50803160635786570329644235 50803160635786570329644235 n ---------------------------- + ---------------------------- 1237940039285380274899124224 618970019642690137449562112 and in Maple notation: 50803160635786570329644235/1237940039285380274899124224+ 50803160635786570329644235/618970019642690137449562112*n ------------------------------ n ----- \ Pi j 96 Theorem , 48, : for n>=, 24, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 1608766753466574727105400775 n 1608766753466574727105400775 ------------------------------ + ----------------------------- 19807040628566084398385987584 39614081257132168796771975168 and in Maple notation: 1608766753466574727105400775/19807040628566084398385987584*n+ 1608766753466574727105400775/39614081257132168796771975168 ------------------------------ n ----- \ Pi j 98 Theorem , 49, : for n>=, 25, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 3184701532372607112841303575 3184701532372607112841303575 n ----------------------------- + ------------------------------ 79228162514264337593543950336 39614081257132168796771975168 and in Maple notation: 3184701532372607112841303575/79228162514264337593543950336+ 3184701532372607112841303575/39614081257132168796771975168*n ------------------------------ n ----- \ Pi j 100 Theorem , 50, : for n>=, 25, we have :, ) sin(-------) = / 2 n + 1 ----- j = 1 12611418068195524166851562157 12611418068195524166851562157 n ------------------------------ + ------------------------------- 316912650057057350374175801344 158456325028528675187087900672 and in Maple notation: 12611418068195524166851562157/316912650057057350374175801344+ 12611418068195524166851562157/158456325028528675187087900672*n ------------------------------ This ends this book that took, 0.733, seconds to generate