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%I #22 Aug 15 2022 02:07:21
%S 2,10,50,58,106,226,250,290,346,514,562,842,866,914
%N Even indices of Pell numbers of the form x^2 + y^2 where x, y > 0.
%C Sequence focuses on the even Pell numbers.
%C Corresponding Pell numbers are 2, 2378, 4866752642924153522, 5616228332641321147898, 13264095873479197467931567359068050319018, ...
%C First differences are 8, 40, 8, 48, 120, 24, 40, ...
%e 2 is a term since A000129(2) = 2 = 1^2 + 1^2.
%e 10 is a term since A000129(10) = 2378 = 13^2 + 47^2.
%e 50 is a term since A000129(50) = 4866752642924153522 = 101254909^2 + 2203746829^2.
%o (PARI) a000129(n) = sum(k=0, n, binomial(n, 2*k+1) * 2^k);
%o isA000404(n) = {for(i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
%o for(n=1, 1e3, if(isA000404(a000129(2*n)), print1(2*n, ", ")));
%Y Cf. A000129, A000404, A001542.
%K nonn,more
%O 1,1
%A _Altug Alkan_, Jan 26 2016
%E a(9)-a(14) from _Jinyuan Wang_, Aug 14 2022