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

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 18, 26, 29, 34, 36, 37, 58, 68, 89, 90, 97, 144, 145, 146, 149, 157, 178, 233, 234, 241, 288, 377, 466, 521, 610, 612, 613, 754, 1000, 1021, 1042, 1076, 1220, 1597, 1600, 1602, 1618, 1741, 2592, 2597, 2605, 2817, 3194, 4181, 4194, 4325, 6770, 6773, 6778, 6786

Offset: 1

Author

Robert Israel, Apr 14 2023

Keywords

• nonn

Comments

Intersection of A001481 and A084176.

Examples

			a(5) = 5 is a term because 5 = 2 + 3 = A000045(3) + A000045(4) = 2^2 + 1^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A000045, A001481, A059389, A084176, A111378.

Programs

Maple

ss:= proc(n) local F,t;
  F:= ifactors(n)[2];
  andmap(t ->  t[1] mod 4 <> 3 or t[2]::even, F)
end proc:
fibs:= map(combinat:-fibonacci, {$0..25}):
N:= max(fibs):
fib2:= {seq(seq(fibs[i]+fibs[j],i=1..j),j=1..nops(fibs))}:
sort(convert(select(t -> t <= N and ss(t), fib2),list));

Mathematica

max = 150; (* max = 150 gives 1670 terms *)
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 *)

Python

from itertools import islice
from sympy import factorint
def A362295_gen(): # generator of terms
    yield from (0,1,2)
    a = [1,2]
    while True:
        b = a[-1]+a[-2]
        c = a[-1]<<1
        flag = True
        for d in a:
            n = b+d
            if flag and n>=c:
                if n>c:
                   f = factorint(c)
                   if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                       yield c
                flag = False
            f = factorint(n)
            if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                yield n
        a.append(b)
A362295_list = list(islice(A362295_gen(),60)) # Chai Wah Wu, Apr 16 2023