A199657
Numerators of lower rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.
Original entry on oeis.org
25, 333, 1667438, 9252915567, 136727214560643, 4607472064276325091, 281395884679127288508771, 31300458157678523147391901818, 3630416277654441522583270655032758, 631040767628866632706111841438119582182, 355477406146830706663807382201012685829049871, 215421112450033407479085892668138597831784081541979
Offset: 1
a(1) = 25 because Kochański's first lower bound was 25/8 = a(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/A199672(1).
a(2) = R(1) * A191642(1) + 3 = 22*15 + 3 = 330 + 3 = 333,
R(2) = R(1) * (A191642(1) + 1 ) + 3 = 22*(15 + 1) + 3 = 355 = A199671(2).
A199658
Denominators of lower rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.
Original entry on oeis.org
8, 106, 530762, 2945294501, 43521624105025, 1466603908374792097, 89571092024800092397857, 9963245273671152951934207006, 1155597392139966274078899403965586, 200866514921276434616104042029044754594, 113151972691506812691685713772827327500605957, 68570669785555705551463950663318228291679702401993
Offset: 1
a(1) = 8 because Kochański's first lower bound was 25/8 = A199657(1)/a(1) and his first upper bound was 22/7 = A199671(1)/A199672(1).
a(2) = S(1) * A191642(1) + 1 = 7*15 + 1 = 105 + 1 = 106,
S(2) = S(1) * (A191642(1) + 1 ) + 1 = 7*(15 + 1) + 1 = 113 = A199672(2).
A199671
Numerators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.
Original entry on oeis.org
22, 355, 1667793, 9254583360, 136736469144003, 4607608800745469094, 281400492287928033977865, 31300739558170811075425879683, 3630447578393999693394346080912441, 631044398076445026705805235784200494623, 355478037191228783108834088006248470029544494, 215421467928070598707869001502226604080254111086473
Offset: 1
a(1) = 22 because Kochański's first lower bound was 25/8 = A199657/A199658(1) and his first upper bound was 22/7 = a(1)/A199672(1).
a(2) = a(1) * (A191642(1) + 1) + 3 = 22*(15 + 1) + 3 = 352 + 3 = 355,
a(3) = a(2) * (A191642(2) + 1) + 3 = 355*(4697 + 1) + 3 = 1667793,
a(4) = a(3) * (A191642(3) + 1) + 3 = 1667793*(5548 + 1) + 3 = 9254583360.
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g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
R = 22; S = 7;
Reap[Print[R]; Sow[R]; For[i = 1, i <= 4, i++, b = g[R, S]; S = S*(b+1)+1; R = R*(b+1)+3; Print[R]; Sow[R]]][[2, 1]] (* Jean-François Alcover, Feb 21 2019 *)
A199672
Denominators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.
Original entry on oeis.org
7, 113, 530875, 2945825376, 43524569930401, 1466647432944722498, 89572558672233037120355, 9963334846229825184971327361, 1155607355474812503904084375292947, 200867670528631909428607946113420047541, 113152173559177341323595142380773440920653498, 68570782937729264728805274258460609065120623055491
Offset: 1
a(1) = 7 because Kochański's first lower bound was 25/8 = A199657(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/a(1).
a(2) = a(1) * (A191642(1) + 1) + 1 = 7*(15 + 1) + 1 = 112 + 1 = 113,
a(3) = a(2) * (A191642(2) + 1) + 1 = 113*(4697 + 1) + 1 = 530875,
a(4) = a(3) * (A191642(3) + 1) + 1 = 530875*(5548 + 1) + 1 = 2945825376.
A221706
Kochanski approximates to sqrt(2) starting with R_0=3, S_0=2.
Original entry on oeis.org
2, 4, 4, 15, 17, 77, 101, 119, 143, 250, 362, 1401, 31168, 88629, 184654, 259251, 298769, 4196069, 38538873, 616984562, 1975413034, 5345718056, 27843871196, 54516286512, 334398528973, 445879679625, 495957494385, 2450869042060, 2629541150528, 4088114099884
Offset: 0
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galpha(alpha, R, S) = {floor((alpha - floor(alpha))/(R - alpha*S));}
fuks() = { n = 29; default(realprecision, 200); alpha = sqrt(2); R = 3; S = 2; x = galpha(alpha, R, S); print1(x, ", "); for (i=1, n, R = R*(x+1) + floor(alpha); S = S*(x+1) + 1; x = galpha(alpha, R, S); print1(x, ", "););}
\\ Michel Marcus, Feb 07 2013
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