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A362295 Sums of two Fibonacci numbers that are also sums of two squares.

%I A362295 #22 Sep 29 2024 06:24:04
%S A362295 0,1,2,4,5,8,9,10,13,16,18,26,29,34,36,37,58,68,89,90,97,144,145,146,
%T A362295 149,157,178,233,234,241,288,377,466,521,610,612,613,754,1000,1021,
%U A362295 1042,1076,1220,1597,1600,1602,1618,1741,2592,2597,2605,2817,3194,4181,4194,4325,6770,6773,6778,6786
%N A362295 Sums of two Fibonacci numbers that are also sums of two squares.
%C A362295 Intersection of A001481 and A084176.
%H A362295 Robert Israel, <a href="/A362295/b362295.txt">Table of n, a(n) for n = 1..2500</a>
%e A362295 a(5) = 5 is a term because 5 = 2 + 3 = A000045(3) + A000045(4) = 2^2 + 1^2.
%p A362295 ss:= proc(n) local F,t;
%p A362295   F:= ifactors(n)[2];
%p A362295   andmap(t ->  t[1] mod 4 <> 3 or t[2]::even, F)
%p A362295 end proc:
%p A362295 fibs:= map(combinat:-fibonacci, {$0..25}):
%p A362295 N:= max(fibs):
%p A362295 fib2:= {seq(seq(fibs[i]+fibs[j],i=1..j),j=1..nops(fibs))}:
%p A362295 sort(convert(select(t -> t <= N and ss(t), fib2),list));
%t A362295 max = 150; (* max = 150 gives 1670 terms *)
%t A362295 Join[{0, 1}, Select[Union[Total /@ Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] && SquaresR[2, #] != 0&]] (* _Jean-François Alcover_, Sep 29 2024, after _Harvey P. Dale_ in A059389 *)
%o A362295 (Python)
%o A362295 from itertools import islice
%o A362295 from sympy import factorint
%o A362295 def A362295_gen(): # generator of terms
%o A362295     yield from (0,1,2)
%o A362295     a = [1,2]
%o A362295     while True:
%o A362295         b = a[-1]+a[-2]
%o A362295         c = a[-1]<<1
%o A362295         flag = True
%o A362295         for d in a:
%o A362295             n = b+d
%o A362295             if flag and n>=c:
%o A362295                 if n>c:
%o A362295                    f = factorint(c)
%o A362295                    if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
%o A362295                        yield c
%o A362295                 flag = False
%o A362295             f = factorint(n)
%o A362295             if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
%o A362295                 yield n
%o A362295         a.append(b)
%o A362295 A362295_list = list(islice(A362295_gen(),60)) # _Chai Wah Wu_, Apr 16 2023
%Y A362295 Cf. A000045, A001481, A059389, A084176, A111378.
%K A362295 nonn
%O A362295 1,3
%A A362295 _Robert Israel_, Apr 14 2023