This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007509 M2061 #54 Feb 16 2025 08:32:31
%S A007509 1,2,13,76,263,2578,36979,33976,622637,11064338,11757173,255865444,
%T A007509 1346255081,3852854518,116752370597,3473755390832,3610501179557,
%U A007509 3481569435902,133330680156299,129049485078524,5457995496252709,227848175409504262,234389556075339277
%N A007509 Numerator of Sum_{k=0..n} (-1)^k/(2*k+1).
%C A007509 Denominators of convergents to 4/Pi. [For Brouncker's continued fraction, with numerators A025547(n+1), for n >= 0. - _Wolfdieter Lang_, Aug 26 2019]
%C A007509 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
%D A007509 P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 131.
%D A007509 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007509 Harvey P. Dale, <a href="/A007509/b007509.txt">Table of n, a(n) for n = 0..1000</a>
%H A007509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pi.html">Pi</a>.
%H A007509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi - Continued Fraction</a>.
%H A007509 R. G. Wilson, V, <a href="/A007509/a007509.pdf">Notes with attachment</a>.
%F A007509 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
%F A007509 a(n) = numerator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - _Vaclav Kotesovec_, May 16 2022
%e A007509 1/1, 2/3, 13/15, 76/105, 263/315, 2578/3465, 36979/45045, 33976/45045, 622637/765765, ...
%p A007509 A007509 := n->numer(add((-1)^k/(2*k+1),k=0..n));
%t A007509 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 *)
%t A007509 Numerator[Table[Sum[(-1)^k/(2k+1),{k,0,n}],{n,0,30}]] (* _Harvey P. Dale_, Oct 22 2011 *)
%t A007509 Table[(-1)^k/(2k+1),{k,0,30}]//Accumulate//Numerator (* _Harvey P. Dale_, May 03 2019 *)
%o A007509 (Magma) [Numerator(&+[(-1)^k/(2*k+1):k in [0..n]]): n in [0..23]]; // _Marius A. Burtea_, Aug 26 2019
%Y A007509 Denominators are given in A352395.
%Y A007509 From _Johannes W. Meijer_, Nov 12 2009: (Start)
%Y A007509 Cf. A157142 and A166107.
%Y A007509 Appears in A167576, A167577, A167578, A024199, A167588 and A167589. (End)
%Y A007509 Cf. A142969 for the numerators of Brouncker's continued fraction of 4/Pi - 1.
%K A007509 nonn,easy,nice,frac
%O A007509 0,2
%A A007509 _N. J. A. Sloane_
%E A007509 Crossref. corrected (A025547 replaced with A352395) by _Wolfdieter Lang_, Apr 06 2022