A060402 -id:A060402 - cp's OEIS frontend

A083124 Continued fraction expansion of tanh(Pi/2).

0, 1, 11, 14, 4, 1, 1, 1, 3, 1, 295, 4, 4, 1, 5, 17, 2, 8, 2, 15, 3, 1, 1, 1, 5, 54, 4, 1, 2, 1, 1, 16, 2, 2, 2, 5, 1, 1, 2, 1, 82, 1, 6, 1, 1, 1, 1, 3, 1, 1, 4, 1, 3, 3, 1, 5, 1, 1, 1, 282, 1, 5, 1, 1, 1, 1, 2, 10, 2, 1, 39, 1, 1, 5, 2, 1, 6, 4, 1, 22, 1, 1, 6, 1, 3, 5, 3, 1, 2, 9, 1, 3, 6, 23, 1, 1, 1, 14, 2

Offset: 0

N. J. A. Sloane, Jun 02 2003

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 11.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A060402, A084304, A367960 (decimal expansion).

with(numtheory): c := cfrac (tanh(Pi/2),300,'quotients');

ContinuedFraction[Tanh[Pi/2], 100] (* Paolo Xausa, Jul 04 2024 *)

contfrac(tanh(Pi/2)) \\ Michel Marcus, Apr 11 2021

A084304 Continued fraction expansion of tanh(Pi^2).

Original entry on oeis.org

0, 1, 186895766, 8, 1, 11, 2, 3, 5, 8, 2, 2, 1, 19, 2, 23, 3, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 25, 1, 19, 1, 1, 1, 6, 6, 2, 11, 1, 5, 2, 11, 1, 3, 1, 2, 1, 5, 1, 2, 7, 15, 4, 1, 7, 24, 1, 1, 1, 18, 2, 2, 25, 6, 4, 1, 1, 1, 2, 1, 2, 2, 2, 40, 1, 3, 3, 2, 2, 2, 16, 1, 361, 1, 2, 1, 4, 1, 13, 1, 5, 2, 1, 5

Offset: 0

Labos Elemer, Jun 04 2003

The least term that is larger than a(2) is a(6427449) = 890522226. - Amiram Eldar, Mar 08 2025

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A060402, A083124.

with(numtheory): c := cfrac (tanh(Pi^2), 300, 'quotients');

ContinuedFraction[Tanh[Pi^2],120] (* Harvey P. Dale, Nov 10 2021 *)

contfrac(tanh(Pi^2)) \\ Michel Marcus, Apr 05 2015

A190352 The continued fraction expansion of tanh(Pi) requires the computation of the pairs (p_n, q_n); sequence gives values of q_n.

Original entry on oeis.org

1, 1, 268, 1073, 15290, 16363, 48016, 64379, 176774, 417927, 594701, 1607329, 5416688, 44940833, 140239187, 185180020, 1066139287, 4449737168, 5515876455, 81672007538, 822235951835, 903907959373, 18900395139295, 719118923252583, 738019318391878

Offset: 0

N. J. A. Sloane, May 09 2011

nonn

a(2) = 268 explains the comment in A021085 that "The decimal expansion of Sum_{n>=1} floor(n * tanh(Pi))/10^n is the same as that of 1/81 for the first 268 decimal places [Borwein et al.]".

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 13.

G. C. Greubel, Table of n, a(n) for n = 0..1920

Cf. A060402, A021085.

lim:=50: with(numtheory): cfr := cfrac(tanh(Pi),lim+10,'quotients'): q[0]:=1:q[1]:=cfr[2]: printf("%d, %d, ", q[0], q[1]): for n from 2 to lim do q[n]:=cfr[n+1]*q[n-1]+q[n-2]: printf("%d, ",q[n]): od: # Nathaniel Johnston, May 10 2011

a[0] := 1; a[1] := 1; A060402:= ContinuedFraction[Tanh[Pi], 100];
a[n_]:= a[n] = A060402[[n + 1]]*a[n - 1] + a[n - 2]; Join[{1, 1}, Table[a[n], {n, 2, 75}]] (* G. C. Greubel, Apr 05 2018 *)

a(n) = A060402(n)*a(n-1) + a(n-2) for n >= 2. - Nathaniel Johnston, May 10 2011

a(4)-a(24) from Nathaniel Johnston, May 10 2011

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A083124 Continued fraction expansion of tanh(Pi/2).

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A084304 Continued fraction expansion of tanh(Pi^2).

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PARI

A190352 The continued fraction expansion of tanh(Pi) requires the computation of the pairs (p_n, q_n); sequence gives values of q_n.

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