A214852 Indices of Fibonacci numbers with the same number of 1's and 0's in their binary representation.

3, 36, 42, 59, 116, 156, 168, 211, 237, 246, 280, 335, 355, 399, 404, 416, 433, 442, 569, 580, 652, 698, 761, 770, 865, 897, 940, 989, 1041, 1049, 1101, 1144, 1214, 1286, 1335, 1352, 1369, 1395, 1698, 1726, 1810, 1928, 1940, 1951, 2055, 2159, 2326, 2332

Offset: 1

Author

Alex Ratushnyak, Mar 08 2013

Keywords

• nonn
• base

Comments

Conjecture: the sequence is infinite.

The sequence of Fibonacci numbers with the same number of 1's and 0's in their binary representation begins: 2, 14930352, 267914296, ... = A259407.

Examples

			Fibonacci(36) = 14930352 = 111000111101000110110000_2, twelve 1's and twelve 0's, therefore 36 is in the sequence.
		

Links

• David Radcliffe, Table of n, a(n) for n = 1..10000 (terms 1..400 from T. D. Noe).

Crossrefs

Cf. A000045, A004685, A259407.

Programs

Mathematica

fQ[n_] := Module[{f = IntegerDigits[Fibonacci[n], 2]}, Count[f, 0] == Count[f, 1]]; Select[Range[3000], fQ] (* T. D. Noe, Mar 08 2013 *)
Position[Fibonacci[Range[2500]],?(DigitCount[#,2,1]==DigitCount[#,2,0]&)]//Flatten (* _Harvey P. Dale, Aug 17 2025 *)

Python

def count10(x):
    c0, c1, m = 0, 0, 1
    while m<=x:
      if x&m:
        c1+=1
      else:
        c0+=1
      m+=m
    return c0-c1
prpr, prev = 0,1
TOP = 3000
for i in range(1,TOP):
    if count10(prev)==0:
        print(i, end=", ")
    prpr, prev = prev, prpr+prev

Python

from sympy import fibonacci
print([n for n in range(3000) if (f := bin(fibonacci(n))[2:]).count('0') == f.count('1')]) # David Radcliffe, May 31 2025