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

1,1

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000045, A004685, A259407.

Sequence in context: A389620 A158301 A105758 * A113799 A072682 A158207

Adjacent sequences: A214849 A214850 A214851 * A214853 A214854 A214855

Keyword

nonn,base

Author

Alex Ratushnyak, Mar 08 2013

Status

approved