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 A157142 #37 Nov 21 2022 17:13:13
%S A157142 1,-3,5,-7,9,-11,13,-15,17,-19,21,-23,25,-27,29,-31,33,-35,37,-39,41,
%T A157142 -43,45,-47,49,-51,53,-55,57,-59,61,-63,65,-67,69,-71,73,-75,77,-79,
%U A157142 81,-83,85,-87,89,-91,93,-95,97,-99,101,-103,105,-107,109,-111,113,-115
%N A157142 Signed denominators of Leibniz series for Pi/4.
%C A157142 Numerators are all 1.
%C A157142 a(n) is also the determinant of the n X n matrix with 1's on the diagonal and 2's elsewhere (cf. A000354). - Jody Nagel (SejeongY(AT)aol.com), May 01 2010
%H A157142 Vincenzo Librandi, <a href="/A157142/b157142.txt">Table of n, a(n) for n = 0..1000</a>
%H A157142 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/pi.html">Archimedes' constant</a>
%H A157142 Mathpages, <a href="https://www.mathpages.com/home/kmath477/kmath477.htm">How Leibniz might have anticipated Euler</a>
%H A157142 Wikipedia, <a href="http://en.wikipedia.org/wiki/Leibniz_formula_for_pi">Leibniz formula for Pi</a>
%H A157142 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1).
%F A157142 Euler transform of length 2 sequence [-3, 2]. - _Michael Somos_, Mar 26 2011
%F A157142 a(n) = b(2*n + 1) where b(n) is completely multiplicative with b(2) = 0, b(p) = p if p == 1 (mod 4), b(p) = -p if p == 3 (mod 4). - _Michael Somos_, Mar 26 2011
%F A157142 With offset 1 this sequence is the exponential reversion of A005264. - _Michael Somos_, Mar 26 2011
%F A157142 a(-1 - n) = a(n), a(n + 1) + a(n - 1) = -2*a(n) for all n in Z. - _Michael Somos_, Mar 26 2011
%F A157142 E.g.f.: (1 - 2*x)*exp(-x). - _Michael Somos_, Mar 26 2011
%F A157142 a(n) = A005408(n)*A033999(n).
%F A157142 G.f.: (1 - x)/(1 + x)^2 = (1 - x)^3 / (1 - x^2)^2.
%F A157142 a(0) = 1, a(1) = -3, a(n) = -2*a(n-1) - a(n-2) for n >= 2.
%F A157142 Sum_{n=0..inf} 1/a(n) = Pi/4.
%e A157142 G.f. = 1 - 3*x + 5*x^2 - 7*x^3 + 9*x^4 - 11*x^5 + 13*x^6 - 15*x^7 + 17*x^8 + ...
%t A157142 a[ n_] := (2*n + 1) * (-1)^n; (* _Michael Somos_, Nov 21 2022 *)
%o A157142 (PARI) {a(n) = (2*n + 1) * (-1)^n};
%o A157142 (Magma) [(2*n + 1) * (-1)^n: n in [0..70]]; // _Vincenzo Librandi_, Dec 23 2018
%Y A157142 Cf. A005264, A005408, A033999.
%Y A157142 Cf. A157327. [_Jaume Oliver Lafont_, Mar 03 2009]
%K A157142 frac,sign,easy
%O A157142 0,2
%A A157142 _Jaume Oliver Lafont_, Feb 24 2009