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

%I A214852 #35 Aug 17 2025 13:55:25
%S A214852 3,36,42,59,116,156,168,211,237,246,280,335,355,399,404,416,433,442,
%T A214852 569,580,652,698,761,770,865,897,940,989,1041,1049,1101,1144,1214,
%U A214852 1286,1335,1352,1369,1395,1698,1726,1810,1928,1940,1951,2055,2159,2326,2332
%N A214852 Indices of Fibonacci numbers with the same number of 1's and 0's in their binary representation.
%C A214852 Conjecture: the sequence is infinite.
%C A214852 The sequence of Fibonacci numbers with the same number of 1's and 0's in their binary representation begins: 2, 14930352, 267914296, ... = A259407.
%H A214852 David Radcliffe, <a href="/A214852/b214852.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..400 from T. D. Noe).
%e A214852 Fibonacci(36) = 14930352 = 111000111101000110110000_2, twelve 1's and twelve 0's, therefore 36 is in the sequence.
%t A214852 fQ[n_] := Module[{f = IntegerDigits[Fibonacci[n], 2]}, Count[f, 0] == Count[f, 1]]; Select[Range[3000], fQ] (* _T. D. Noe_, Mar 08 2013 *)
%t A214852 Position[Fibonacci[Range[2500]],_?(DigitCount[#,2,1]==DigitCount[#,2,0]&)]//Flatten (* _Harvey P. Dale_, Aug 17 2025 *)
%o A214852 (Python)
%o A214852 def count10(x):
%o A214852     c0, c1, m = 0, 0, 1
%o A214852     while m<=x:
%o A214852       if x&m:
%o A214852         c1+=1
%o A214852       else:
%o A214852         c0+=1
%o A214852       m+=m
%o A214852     return c0-c1
%o A214852 prpr, prev = 0,1
%o A214852 TOP = 3000
%o A214852 for i in range(1,TOP):
%o A214852     if count10(prev)==0:
%o A214852         print(i, end=", ")
%o A214852     prpr, prev = prev, prpr+prev
%o A214852 (Python)
%o A214852 from sympy import fibonacci
%o A214852 print([n for n in range(3000) if (f := bin(fibonacci(n))[2:]).count('0') == f.count('1')]) # _David Radcliffe_, May 31 2025
%Y A214852 Cf. A000045, A004685, A259407.
%K A214852 nonn,base
%O A214852 1,1
%A A214852 _Alex Ratushnyak_, Mar 08 2013