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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
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
KEYWORD
nonn,easy,nice,frac
AUTHOR
EXTENSIONS
Crossref. corrected (A025547 replaced with A352395) by Wolfdieter Lang, Apr 06 2022
STATUS
approved

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Last modified February 27 10:59 EST 2024. Contains 370378 sequences. (Running on oeis4.)