n ----- \ Pi (2 j - 1) (2 i) evaluations of , ) sin(------------) , for i from 1 to , 50 / 4 n + 2 ----- j = 1 By Shalosh B. Ekhad Comment: It can be easily shown that for each positive integer i n ----- \ Pi (2 j - 1) (2 i) binomial(2 i, i) n binomial(2 i - 1, i) ) sin(------------) = ------------------ + -------------------- - 1/2 / 4 n + 2 i i ----- 4 4 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 - 1/4 / 4 n + 2 ----- j = 1 and in Maple notation: 1/2*n-1/4 ------------------------------ n ----- \ Pi (2 j - 1) 4 3 n Theorem , 2, : for n>=, 2, we have :, ) sin(------------) = --- - 5/16 / 4 n + 2 8 ----- j = 1 and in Maple notation: 3/8*n-5/16 ------------------------------ n ----- \ Pi (2 j - 1) 6 5 n 11 Theorem , 3, : for n>=, 2, we have :, ) sin(------------) = --- - -- / 4 n + 2 16 32 ----- j = 1 and in Maple notation: 5/16*n-11/32 ------------------------------ n ----- \ Pi (2 j - 1) 8 35 n 93 Theorem , 4, : for n>=, 3, we have :, ) sin(------------) = ---- - --- / 4 n + 2 128 256 ----- j = 1 and in Maple notation: 35/128*n-93/256 ------------------------------ n ----- \ Pi (2 j - 1) 10 63 n 193 Theorem , 5, : for n>=, 3, we have :, ) sin(------------) = ---- - --- / 4 n + 2 256 512 ----- j = 1 and in Maple notation: 63/256*n-193/512 ------------------------------ Theorem , 6, : for n>=, 4, we have :, n ----- \ Pi (2 j - 1) 12 793 231 n ) sin(------------) = - ---- + ----- / 4 n + 2 2048 1024 ----- j = 1 and in Maple notation: -793/2048+231/1024*n ------------------------------ Theorem , 7, : for n>=, 4, we have :, n ----- \ Pi (2 j - 1) 14 1619 429 n ) sin(------------) = - ---- + ----- / 4 n + 2 4096 2048 ----- j = 1 and in Maple notation: -1619/4096+429/2048*n ------------------------------ Theorem , 8, : for n>=, 5, we have :, n ----- \ Pi (2 j - 1) 16 26333 6435 n ) sin(------------) = - ----- + ------ / 4 n + 2 65536 32768 ----- j = 1 and in Maple notation: -26333/65536+6435/32768*n ------------------------------ Theorem , 9, : for n>=, 5, we have :, n ----- \ Pi (2 j - 1) 18 53381 12155 n ) sin(------------) = - ------ + ------- / 4 n + 2 131072 65536 ----- j = 1 and in Maple notation: -53381/131072+12155/65536*n ------------------------------ Theorem , 10, : for n>=, 6, we have :, n ----- \ Pi (2 j - 1) 20 215955 46189 n ) sin(------------) = - ------ + ------- / 4 n + 2 524288 262144 ----- j = 1 and in Maple notation: -215955/524288+46189/262144*n ------------------------------ Theorem , 11, : for n>=, 6, we have :, n ----- \ Pi (2 j - 1) 22 436109 88179 n ) sin(------------) = - ------- + ------- / 4 n + 2 1048576 524288 ----- j = 1 and in Maple notation: -436109/1048576+88179/524288*n ------------------------------ Theorem , 12, : for n>=, 7, we have :, n ----- \ Pi (2 j - 1) 24 3518265 676039 n ) sin(------------) = - ------- + -------- / 4 n + 2 8388608 4194304 ----- j = 1 and in Maple notation: -3518265/8388608+676039/4194304*n ------------------------------ Theorem , 13, : for n>=, 7, we have :, n ----- \ Pi (2 j - 1) 26 7088533 1300075 n ) sin(------------) = - -------- + --------- / 4 n + 2 16777216 8388608 ----- j = 1 and in Maple notation: -7088533/16777216+1300075/8388608*n ------------------------------ Theorem , 14, : for n>=, 8, we have :, n ----- \ Pi (2 j - 1) 28 28539857 5014575 n ) sin(------------) = - -------- + --------- / 4 n + 2 67108864 33554432 ----- j = 1 and in Maple notation: -28539857/67108864+5014575/33554432*n ------------------------------ Theorem , 15, : for n>=, 8, we have :, n ----- \ Pi (2 j - 1) 30 57414019 9694845 n ) sin(------------) = - --------- + --------- / 4 n + 2 134217728 67108864 ----- j = 1 and in Maple notation: -57414019/134217728+9694845/67108864*n ------------------------------ Theorem , 16, : for n>=, 9, we have :, n ----- \ Pi (2 j - 1) 32 1846943453 300540195 n ) sin(------------) = - ---------- + ----------- / 4 n + 2 4294967296 2147483648 ----- j = 1 and in Maple notation: -1846943453/4294967296+300540195/2147483648*n ------------------------------ Theorem , 17, : for n>=, 9, we have :, n ----- \ Pi (2 j - 1) 34 3711565741 583401555 n ) sin(------------) = - ---------- + ----------- / 4 n + 2 8589934592 4294967296 ----- j = 1 and in Maple notation: -3711565741/8589934592+583401555/4294967296*n ------------------------------ Theorem , 18, : for n>=, 10, we have :, n ----- \ Pi (2 j - 1) 36 14911085359 2268783825 n ) sin(------------) = - ----------- + ------------ / 4 n + 2 34359738368 17179869184 ----- j = 1 and in Maple notation: -14911085359/34359738368+2268783825/17179869184*n ------------------------------ Theorem , 19, : for n>=, 10, we have :, n ----- \ Pi (2 j - 1) 38 29941580393 4418157975 n ) sin(------------) = - ----------- + ------------ / 4 n + 2 68719476736 34359738368 ----- j = 1 and in Maple notation: -29941580393/68719476736+4418157975/34359738368*n ------------------------------ Theorem , 20, : for n>=, 11, we have :, n ----- \ Pi (2 j - 1) 40 240416274739 34461632205 n ) sin(------------) = - ------------ + ------------- / 4 n + 2 549755813888 274877906944 ----- j = 1 and in Maple notation: -240416274739/549755813888+34461632205/274877906944*n ------------------------------ Theorem , 21, : for n>=, 11, we have :, n ----- \ Pi (2 j - 1) 42 482473579583 67282234305 n ) sin(------------) = - ------------- + ------------- / 4 n + 2 1099511627776 549755813888 ----- j = 1 and in Maple notation: -482473579583/1099511627776+67282234305/549755813888*n ------------------------------ Theorem , 22, : for n>=, 12, we have :, n ----- \ Pi (2 j - 1) 44 1936010885087 263012370465 n ) sin(------------) = - ------------- + -------------- / 4 n + 2 4398046511104 2199023255552 ----- j = 1 and in Maple notation: -1936010885087/4398046511104+263012370465/2199023255552*n ------------------------------ Theorem , 23, : for n>=, 12, we have :, n ----- \ Pi (2 j - 1) 46 3883457090629 514589420475 n ) sin(------------) = - ------------- + -------------- / 4 n + 2 8796093022208 4398046511104 ----- j = 1 and in Maple notation: -3883457090629/8796093022208+514589420475/4398046511104*n ------------------------------ Theorem , 24, : for n>=, 13, we have :, n ----- \ Pi (2 j - 1) 48 62306843256889 8061900920775 n ) sin(------------) = - --------------- + --------------- / 4 n + 2 140737488355328 70368744177664 ----- j = 1 and in Maple notation: -62306843256889/140737488355328+8061900920775/70368744177664*n ------------------------------ Theorem , 25, : for n>=, 13, we have :, n ----- \ Pi (2 j - 1) 50 124936162550609 15801325804719 n ) sin(------------) = - --------------- + ---------------- / 4 n + 2 281474976710656 140737488355328 ----- j = 1 and in Maple notation: -124936162550609/281474976710656+15801325804719/140737488355328*n ------------------------------ Theorem , 26, : for n>=, 14, we have :, n ----- \ Pi (2 j - 1) 52 500960136802799 61989816618513 n ) sin(------------) = - ---------------- + ---------------- / 4 n + 2 1125899906842624 562949953421312 ----- j = 1 and in Maple notation: -500960136802799/1125899906842624+61989816618513/562949953421312*n ------------------------------ Theorem , 27, : for n>=, 14, we have :, n ----- \ Pi (2 j - 1) 54 1004216192739617 121683714103007 n ) sin(------------) = - ---------------- + ----------------- / 4 n + 2 2251799813685248 1125899906842624 ----- j = 1 and in Maple notation: -1004216192739617/2251799813685248+121683714103007/1125899906842624*n ------------------------------ Theorem , 28, : for n>=, 15, we have :, n ----- \ Pi (2 j - 1) 56 8051112929645937 956086325095055 n ) sin(------------) = - ----------------- + ----------------- / 4 n + 2 18014398509481984 9007199254740992 ----- j = 1 and in Maple notation: -8051112929645937/18014398509481984+956086325095055/9007199254740992*n ------------------------------ Theorem , 29, : for n>=, 15, we have :, n ----- \ Pi (2 j - 1) 58 16135194353260669 1879204156221315 n ) sin(------------) = - ----------------- + ------------------ / 4 n + 2 36028797018963968 18014398509481984 ----- j = 1 and in Maple notation: -16135194353260669/36028797018963968+1879204156221315/18014398509481984*n ------------------------------ Theorem , 30, : for n>=, 16, we have :, n ----- \ Pi (2 j - 1) 60 64666057690124097 7391536347803839 n ) sin(------------) = - ------------------ + ------------------ / 4 n + 2 144115188075855872 72057594037927936 ----- j = 1 and in Maple notation: -64666057690124097/144115188075855872+7391536347803839/72057594037927936*n ------------------------------ Theorem , 31, : for n>=, 16, we have :, n ----- \ Pi (2 j - 1) 62 129570552036628963 14544636039226909 n ) sin(------------) = - ------------------ + ------------------- / 4 n + 2 288230376151711744 144115188075855872 ----- j = 1 and in Maple notation: -129570552036628963/288230376151711744+14544636039226909/144115188075855872*n ------------------------------ Theorem , 32, : for n>=, 17, we have :, n ----- \ Pi (2 j - 1) 64 8307059966383480541 916312070471295267 n ) sin(------------) = - -------------------- + -------------------- / 4 n + 2 18446744073709551616 9223372036854775808 ----- j = 1 and in Maple notation: -8307059966383480541/18446744073709551616+916312070471295267/ 9223372036854775808*n ------------------------------ Theorem , 33, : for n>=, 17, we have :, n ----- \ Pi (2 j - 1) 66 16641886965205485181 1804857108504066435 n ) sin(------------) = - -------------------- + --------------------- / 4 n + 2 36893488147419103232 18446744073709551616 ----- j = 1 and in Maple notation: -16641886965205485181/36893488147419103232+1804857108504066435/ 18446744073709551616*n ------------------------------ Theorem , 34, : for n>=, 18, we have :, n ----- \ Pi (2 j - 1) 68 7113260368810144185 n 66673715926028062279 ) sin(------------) = --------------------- - --------------------- / 4 n + 2 73786976294838206464 147573952589676412928 ----- j = 1 and in Maple notation: 7113260368810144185/73786976294838206464*n-66673715926028062279/ 147573952589676412928 ------------------------------ n ----- \ Pi (2 j - 1) 70 Theorem , 35, : for n>=, 18, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 133550667862593557249 14023284727082855679 n - --------------------- + ---------------------- 295147905179352825856 147573952589676412928 and in Maple notation: -133550667862593557249/295147905179352825856+14023284727082855679/ 147573952589676412928*n ------------------------------ n ----- \ Pi (2 j - 1) 72 Theorem , 36, : for n>=, 19, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 1069963485648202108623 110628135069209194801 n - ---------------------- + ----------------------- 2361183241434822606848 1180591620717411303424 and in Maple notation: -1069963485648202108623/2361183241434822606848+110628135069209194801/ 1180591620717411303424*n ------------------------------ n ----- \ Pi (2 j - 1) 74 Theorem , 37, : for n>=, 19, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 2142916920892869330619 218266320541953276229 n - ---------------------- + ----------------------- 4722366482869645213696 2361183241434822606848 and in Maple notation: -2142916920892869330619/4722366482869645213696+218266320541953276229/ 2361183241434822606848*n ------------------------------ n ----- \ Pi (2 j - 1) 76 Theorem , 38, : for n>=, 20, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 8583155384652632758067 861577581086657669325 n - ----------------------- + ----------------------- 18889465931478580854784 9444732965739290427392 and in Maple notation: -8583155384652632758067/18889465931478580854784+861577581086657669325/ 9444732965739290427392*n ------------------------------ n ----- \ Pi (2 j - 1) 78 Theorem , 39, : for n>=, 20, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 17188402502153641353809 1701063429324939500975 n - ----------------------- + ------------------------ 37778931862957161709568 18889465931478580854784 and in Maple notation: -17188402502153641353809/37778931862957161709568+1701063429324939500975/ 18889465931478580854784*n ------------------------------ n ----- \ Pi (2 j - 1) 80 Theorem , 40, : for n>=, 21, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 275354652720323249561139 26876802183334044115405 n - ------------------------ + ------------------------- 604462909807314587353088 302231454903657293676544 and in Maple notation: -275354652720323249561139/604462909807314587353088+26876802183334044115405/ 302231454903657293676544*n ------------------------------ n ----- \ Pi (2 j - 1) 82 Theorem , 41, : for n>=, 21, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 551364837201215622149483 53098072606098965203605 n - ------------------------- + ------------------------- 1208925819614629174706176 604462909807314587353088 and in Maple notation: -551364837201215622149483/1208925819614629174706176+53098072606098965203605/ 604462909807314587353088*n ------------------------------ n ----- \ Pi (2 j - 1) 84 Theorem , 42, : for n>=, 22, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 2207987828452771963131437 209863810776486386280915 n - ------------------------- + -------------------------- 4835703278458516698824704 2417851639229258349412352 and in Maple notation: -2207987828452771963131437/4835703278458516698824704+209863810776486386280915/ 2417851639229258349412352*n ------------------------------ n ----- \ Pi (2 j - 1) 86 Theorem , 43, : for n>=, 22, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 4420856210644531981757779 414847067813984717066925 n - ------------------------- + -------------------------- 9671406556917033397649408 4835703278458516698824704 and in Maple notation: -4420856210644531981757779/9671406556917033397649408+414847067813984717066925/ 4835703278458516698824704*n ------------------------------ n ----- \ Pi (2 j - 1) 88 Theorem , 44, : for n>=, 23, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 35404563054957527191977407 3281063172710606398620225 n - -------------------------- + --------------------------- 77371252455336267181195264 38685626227668133590597632 and in Maple notation: -35404563054957527191977407/77371252455336267181195264+ 3281063172710606398620225/38685626227668133590597632*n ------------------------------ n ----- \ Pi (2 j - 1) 90 Theorem , 45, : for n>=, 23, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 70882038624864178970590819 6489213830472088210604445 n - --------------------------- + --------------------------- 154742504910672534362390528 77371252455336267181195264 and in Maple notation: -70882038624864178970590819/154742504910672534362390528+ 6489213830472088210604445/77371252455336267181195264*n ------------------------------ n ----- \ Pi (2 j - 1) 92 Theorem , 46, : for n>=, 24, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 283810294231216371891519991 25674715590128696833261065 n - --------------------------- + ---------------------------- 618970019642690137449562112 309485009821345068724781056 and in Maple notation: -283810294231216371891519991/618970019642690137449562112+ 25674715590128696833261065/309485009821345068724781056*n ------------------------------ n ----- \ Pi (2 j - 1) 94 Theorem , 47, : for n>=, 24, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 568166859006903567119917877 50803160635786570329644235 n - ---------------------------- + ---------------------------- 1237940039285380274899124224 618970019642690137449562112 and in Maple notation: -568166859006903567119917877/1237940039285380274899124224+ 50803160635786570329644235/618970019642690137449562112*n ------------------------------ n ----- \ Pi (2 j - 1) 96 Theorem , 48, : for n>=, 25, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 18198273875099509671280586809 1608766753466574727105400775 n - ----------------------------- + ------------------------------ 39614081257132168796771975168 19807040628566084398385987584 and in Maple notation: -18198273875099509671280586809/39614081257132168796771975168+ 1608766753466574727105400775/19807040628566084398385987584*n ------------------------------ n ----- \ Pi (2 j - 1) 98 Theorem , 49, : for n>=, 25, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 36429379724759561683930671593 3184701532372607112841303575 n - ----------------------------- + ------------------------------ 79228162514264337593543950336 39614081257132168796771975168 and in Maple notation: -36429379724759561683930671593/79228162514264337593543950336+ 3184701532372607112841303575/39614081257132168796771975168*n ------------------------------ n ----- \ Pi (2 j - 1) 100 Theorem , 50, : for n>=, 26, we have :, ) sin(------------) = / 4 n + 2 ----- j = 1 145844906960333151020236338515 12611418068195524166851562157 n - ------------------------------ + ------------------------------- 316912650057057350374175801344 158456325028528675187087900672 and in Maple notation: -145844906960333151020236338515/316912650057057350374175801344+ 12611418068195524166851562157/158456325028528675187087900672*n ------------------------------ This ends this book that took, 0.953, seconds to generate ----------------- This took, 1.021, seconds.