A007509 Numerator of Sum_{k=0..n} (-1)^k/(2*k+1).

1, 2, 13, 76, 263, 2578, 36979, 33976, 622637, 11064338, 11757173, 255865444, 1346255081, 3852854518, 116752370597, 3473755390832, 3610501179557, 3481569435902, 133330680156299, 129049485078524, 5457995496252709, 227848175409504262, 234389556075339277

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• frac

Comments

Denominators of convergents to 4/Pi. [For Brouncker's continued fraction, with numerators A025547(n+1), for n >= 0. - Wolfdieter Lang, Aug 26 2019]

See A352395 (the denominators for the present sequence) for the formula of this alternating sum, and the Abramowitz-Stegun link. - Wolfdieter Lang, Apr 06 2022

Examples

			1/1, 2/3, 13/15, 76/105, 263/315, 2578/3465, 36979/45045, 33976/45045, 622637/765765, ...
		

References

• P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 131.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pi.
• Eric Weisstein's World of Mathematics, Pi - Continued Fraction.
• R. G. Wilson, V, Notes with attachment.

Crossrefs

Denominators are given in A352395.

From Johannes W. Meijer, Nov 12 2009: (Start)

Cf. A157142 and A166107.

Appears in A167576, A167577, A167578, A024199, A167588 and A167589. (End)

Cf. A142969 for the numerators of Brouncker's continued fraction of 4/Pi - 1.

Programs

Magma

[Numerator(&+[(-1)^k/(2*k+1):k in [0..n]]): n in [0..23]]; // Marius A. Burtea, Aug 26 2019

Maple

A007509 := n->numer(add((-1)^k/(2*k+1),k=0..n));

Mathematica

Table[Numerator[FunctionExpand[(Pi + (-1)^n(HarmonicNumber[n/2 + 1/4] - HarmonicNumber[n/2 - 1/4]))/4]], {n, 0, 20}] (* Vladimir Reshetnikov, Jan 18 2011 *)
Numerator[Table[Sum[(-1)^k/(2k+1),{k,0,n}],{n,0,30}]] (* Harvey P. Dale, Oct 22 2011 *)
Table[(-1)^k/(2k+1),{k,0,30}]//Accumulate//Numerator (* Harvey P. Dale, May 03 2019 *)

Formula

a(n) = numerator((Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), with the digamma function Psi(z). See the formula in A352395. - Wolfdieter Lang, Apr 06 2022

a(n) = numerator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

Extensions

Crossref. corrected (A025547 replaced with A352395) by Wolfdieter Lang, Apr 06 2022