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 A157142 Signed denominators of Leibniz series for Pi/4. 13
 1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29, -31, 33, -35, 37, -39, 41, -43, 45, -47, 49, -51, 53, -55, 57, -59, 61, -63, 65, -67, 69, -71, 73, -75, 77, -79, 81, -83, 85, -87, 89, -91, 93, -95, 97, -99, 101, -103, 105, -107, 109, -111, 113, -115 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Numerators are all 1. 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 X. Gourdon and P. Sebah, Archimedes' constant Mathpages, How Leibniz might have anticipated Euler Wikipedia, Leibniz formula for Pi Index entries for linear recurrences with constant coefficients, signature (-2,-1). FORMULA Euler transform of length 2 sequence [-3, 2]. - Michael Somos, Mar 26 2011 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 With offset 1 this sequence is the exponential reversion of A005264. - Michael Somos, Mar 26 2011 a(-1 - n) = a(n). a(n + 1) + a(n - 1) = -2*a(n). - Michael Somos, Mar 26 2011 E.g.f.: (1 - 2*x)*exp(-x). - Michael Somos, Mar 26 2011 a(n) = A005408(n)*A033999(n). G.f.: (1 - x)/(1 + x)^2. a(0) = 1, a(1) = -3, a(n) = -2*a(n-1) - a(n-2) for n >= 2. Sum_{n=0..inf} 1/a(n) = Pi/4. EXAMPLE 1 - 3*x + 5*x^2 - 7*x^3 + 9*x^4 - 11*x^5 + 13*x^6 - 15*x^7 + 17*x^8 + ... PROG (PARI) {a(n) = (2*n + 1) * (-1)^n} (MAGMA) [(2*n + 1) * (-1)^n: n in [0..70]]; // Vincenzo Librandi, Dec 23 2018 CROSSREFS Cf. A005264, A005408, A033999. Cf. A157327. [Jaume Oliver Lafont, Mar 03 2009] Sequence in context: A081874 A165747 A053229 * A247328 A317107 A317439 Adjacent sequences:  A157139 A157140 A157141 * A157143 A157144 A157145 KEYWORD frac,sign,easy AUTHOR Jaume Oliver Lafont, Feb 24 2009 STATUS approved

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Last modified July 9 14:41 EDT 2020. Contains 335543 sequences. (Running on oeis4.)