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A111766
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Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.
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1
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0, 5, 24, 145, 840, 4901, 28560, 166465, 970224, 5654885, 32959080, 192099601, 1119638520, 6525731525, 38034750624, 221682772225, 1292061882720, 7530688524101, 43892069261880, 255821727047185, 1491038293021224, 8690408031080165, 50651409893459760
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OFFSET
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1,2
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COMMENTS
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This parallels Cassini's identity for Fibonacci numbers (Mathworld).
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).
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FORMULA
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a(n) = A000129(n-1)*A000129(n+1) = A000129(n)^2 + (-1)^n.
G.f. -x^2*(-5+x) / ( (1+x)*(1-6*x+x^2) ). - R. J. Mathar, Sep 21 2011
a(n) = A001333(n)^2 - A000129(n)^2 for n >= 1. - Richard R. Forberg, Aug 24 2013
From Colin Barker, Nov 04 2016: (Start)
a(n) = (6*(-1)^n+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/8 for n>0.
a(n) = 5*a(n-1)+5*a(n-2)-a(n-3) for n>3.
(End)
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EXAMPLE
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a(5) = P(4)*P(6) = 12*70 = 840 = P(5)-1 = 29^2-1.
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PROG
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(PARI) concat(0, Vec(-x^2*(-5+x)/((1+x)*(1-6*x+x^2)) + O(x^30))) \\ Colin Barker, Nov 04 2016
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CROSSREFS
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Cf. A076218, A078522 (bisections).
Sequence in context: A232318 A201952 A221788 * A228067 A322208 A241134
Adjacent sequences: A111763 A111764 A111765 * A111767 A111768 A111769
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KEYWORD
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nonn,easy
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AUTHOR
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Jeremy C. Buchanan (jbuchanan(AT)myhww.org), Nov 21 2005
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STATUS
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approved
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