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