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 A157142 Signed denominators of Leibniz series for Pi/4 12
 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 X. Gourdon and P. Sebah, Archimedes' constant Mathpages, How Leibniz might have anticipated Euler Wikipedia, Leibniz formula for Pi 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)=-2a(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} CROSSREFS Cf. A005264, A005408, A033999. Cf. A157327. [From Jaume Oliver Lafont, Mar 03 2009] Sequence in context: A081874 A165747 A053229 * A247328 A004273 A005408 Adjacent sequences:  A157139 A157140 A157141 * A157143 A157144 A157145 KEYWORD frac,sign AUTHOR Jaume Oliver Lafont, Feb 24 2009 STATUS approved

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