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

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

Author

N. J. A. Sloane, May 09 2011

Keywords

• nonn

Comments

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.]".

References

• 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.

Links

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

Crossrefs

Cf. A060402, A021085.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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