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
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..49
Programs
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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
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Mathematica
With[{f = Fibonacci[Range[0, 200]]}, DeleteDuplicates[Select[f, MemberQ[f, DigitCount[#, 2, 1]] &]]] (* Amiram Eldar, Mar 04 2025 *) -
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
Extensions
a(1) = 0 inserted by Robert Israel, Mar 13 2025