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