login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Even indices of Pell numbers of the form x^2 + y^2 where x, y > 0.
0

%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