A127676 Numerators of partial sums of a series for Pi*sqrt(2)/4.

1, 4, 17, 104, 347, 4132, 50251, 47248, 848261, 16882724, 16189889, 357817912, 1856017421, 5753962988, 161845337077, 4871637351712, 5008383140437, 5137314884092, 185568039683479, 181286844605704, 7599727236867089

Offset: 0

Author

Wolfdieter Lang, Mar 07 2007

Keywords

• nonn
• easy
• frac

Comments

Denominators coincide with A025547(n+1) for n=0..41, but then start to differ. See the W. Lang link. denominator(r(42))=7422822568422519986207785205976075 but the corresponding entry is A025547(43)=126187983663182839765532348501593275.

Examples

			Rationals r(n): [1, 4/3, 17/15, 104/105, 347/315, 4132/3465, ...].
		

References

• E. Maor, Trigonometric Delights, Princeton University Press, 1998, p. 205.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
• W. Lang, Rationals and limit.

Crossrefs

Cf. A025547 (denominators).

Programs

Magma

[Numerator((&+[(-1)^Floor(k/2)/(1+2*k): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 17 2018

Mathematica

Numerator[Table[Sum[(-1)^Floor[k/2]/(2*k + 1), {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Aug 17 2018 *)

PARI

a(n) = numerator(sum(k=0, n, (-1)^(k\2)/(2*k+1))); \\ Michel Marcus, Oct 03 2017

Formula

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = Sum_{k=0..n} (-1)^floor(k/2)/(2*k+1).