A381704 Fibonacci numbers having a Fibonacci number of 1's in their binary representation.

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 144, 233, 987, 4181, 6765, 17711, 832040, 3524578, 1836311903, 2971215073, 225851433717, 259695496911122585, 3928413764606871165730, 26925748508234281076009, 9969216677189303386214405760200, 638817435613190341905763972389505493

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..49

Example

F(10) = (55)_10 = (110111)_2 has five 1's in binary, 5 is a Fibonacci number, thus 55 is a term.
F(12) = (144)_10 = (10010000)_2 has two 1's in binary, 2 is a Fibonacci number, thus 144 is a term.

Maple

isfib:= n -> issqr(5*n^2+4) or issqr(5*n^2-4):
select(n -> isfib(convert(convert(n, base, 2), `+`)), map(combinat:-fibonacci, [0, $2..1000])); # Robert Israel, Mar 13 2025

Mathematica

With[{f = Fibonacci[Range[0, 200]]}, DeleteDuplicates[Select[f, MemberQ[f, DigitCount[#, 2, 1]] &]]] (* Amiram Eldar, Mar 04 2025 *)

Prog

(PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
lista(nn) = for (n=2, nn, my(f = fibonacci(n)); if (isfib(hammingweight(f)), print1(f, ", ")); ); \\ Michel Marcus, Mar 04 2025

Crossrefs

Cf. A000045, A004685, A010056, A382053.

Sequence in context: A147659 A005181 A177376 * A120659 A042581 A349843

Adjacent sequences: A381701 A381702 A381703 * A381705 A381706 A381707

Keyword

nonn,base

Author

Ctibor O. Zizka, Mar 04 2025

Extensions

a(1) = 0 inserted by Robert Israel, Mar 13 2025

Status

approved