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A007509 Numerator of Sum_{k=0..n} (-1)^k/(2*k+1).
(Formerly M2061)
13
1, 2, 13, 76, 263, 2578, 36979, 33976, 622637, 11064338, 11757173, 255865444, 1346255081, 3852854518, 116752370597, 3473755390832, 3610501179557, 3481569435902, 133330680156299, 129049485078524, 5457995496252709, 227848175409504262, 234389556075339277 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

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

EXAMPLE

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

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 *)

PROG

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

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.

Sequence in context: A154357 A161130 A192700 * A077413 A024199 A037523

Adjacent sequences:  A007506 A007507 A007508 * A007510 A007511 A007512

KEYWORD

nonn,easy,nice,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified September 28 01:47 EDT 2022. Contains 357063 sequences. (Running on oeis4.)