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A157142
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Signed denominators of Leibniz series for Pi/4
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9
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Numerators are all 1.
Comment from Jody Nagel (SejeongY(AT)aol.com), May 01 2010: a(n) is also the determinant of the n X n matrix with 1's on the diagonal and 2's elsewhere (cf. A000354).
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LINKS
| X. Gourdon and P. Sebah, Archimedes' constant
Mathpages, How Leibniz might have anticipated Euler
Wikipedia, Leibniz formula for Pi
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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
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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 + ...
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PROG
| (PARI) {a(n) = (2*n + 1) * (-1)^n}
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CROSSREFS
| Cf. A005264, A005408, A033999.
Cf. A157327. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 03 2009]
Sequence in context: A081874 A165747 A053229 * A004273 A005408 A176271
Adjacent sequences: A157139 A157140 A157141 * A157143 A157144 A157145
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KEYWORD
| frac,sign,changed
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AUTHOR
| Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 24 2009
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