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%I #14 Nov 04 2016 05:00:36
%S 0,5,24,145,840,4901,28560,166465,970224,5654885,32959080,192099601,
%T 1119638520,6525731525,38034750624,221682772225,1292061882720,
%U 7530688524101,43892069261880,255821727047185,1491038293021224,8690408031080165,50651409893459760
%N 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.
%C This parallels Cassini's identity for Fibonacci numbers (Mathworld).
%H Colin Barker, <a href="/A111766/b111766.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F a(n) = A000129(n-1)*A000129(n+1) = A000129(n)^2 + (-1)^n.
%F G.f. -x^2*(-5+x) / ( (1+x)*(1-6*x+x^2) ). - _R. J. Mathar_, Sep 21 2011
%F a(n) = A001333(n)^2 - A000129(n)^2 for n >= 1. - _Richard R. Forberg_, Aug 24 2013
%F From _Colin Barker_, Nov 04 2016: (Start)
%F a(n) = (6*(-1)^n+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/8 for n>0.
%F a(n) = 5*a(n-1)+5*a(n-2)-a(n-3) for n>3.
%F (End)
%e a(5) = P(4)*P(6) = 12*70 = 840 = P(5)-1 = 29^2-1.
%o (PARI) concat(0, Vec(-x^2*(-5+x)/((1+x)*(1-6*x+x^2)) + O(x^30))) \\ _Colin Barker_, Nov 04 2016
%Y Cf. A076218, A078522 (bisections).
%K nonn,easy
%O 1,2
%A Jeremy C. Buchanan (jbuchanan(AT)myhww.org), Nov 21 2005
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