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%I #36 Apr 15 2017 08:17:58
%S 2,4,6,12,18,26,36,52,122,244,314,628,794,866,1588,1732,1754
%N Even numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 2*b^2.
%C These Fibonacci numbers F(n) have no prime factor congruent to 5 or 7 mod 8 to an odd power.
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%t Select[Range[2, 200, 2], Length[FindInstance[x^2 + 2*y^2 == Fibonacci[#], {x, y}, Integers]] > 0 &] (* _G. C. Greubel_, Apr 14 2017 *)
%o (PARI) for(i=2, 500, a=factorint(fibonacci(i))~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==0, print(i", ")))
%Y Cf. A000045, A215818, A215819, A215820.
%K nonn,more
%O 1,1
%A _V. Raman_, Aug 23 2012
%E a(12)-a(17) from _V. Raman_, Aug 28 2012