A222296 -id:A222296 - cp's OEIS frontend

A222601 Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.

3, 5, 2, 4, 3, 1, 2, 3, 5, 1, 3, 5, 3, 3, 2, 1, 6, 1, 2, 3, 3, 2, 0, 3, 5, 4, 4, 3, 4, 2, 1, 2, 6, 1, 2, 2, 5, 4, 3, 5, 2, 2, 2, 1, 2, 2, 2, 5, 6, 3, 2, 2, 3, 1, 5, 1, 1, 0, 8, 4, 3, 3, 3, 3, 5, 4, 4, 2, 2, 2, 2, 3, 2, 6, 3, 0, 0, 2, 5, 5, 1, 6, 5, 0, 3, 5, 1

Offset: 0

T. D. Noe, Mar 08 2013

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004685 (Fibonacci numbers in binary), A214853 (one 0-bit), A222602.

Cf. A222295, A222296.

f = Fibonacci[Range[0,100]]; Table[Length[Select[f, Count[IntegerDigits[#, 2], 0] == n &]], {n, 0, 20}]

A222602 Irregular triangle of conjectured Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

1, 1, 3, 0, 2, 5, 13, 55, 21, 987, 8, 89, 233, 377, 34, 1597, 28657, 6765, 144, 610, 17711, 196418, 514229, 2584, 4181, 10946, 121393, 317811, 3524578, 46368, 1346269, 1836311903, 75025, 5702887, 24157817, 102334155, 165580141, 832040, 14930352, 701408733

Offset: 0

T. D. Noe, Mar 08 2013

			The irregular triangle begins
{1, 1, 3},
{0, 2, 5, 13, 55},
{21, 987},
{8, 89, 233, 377},
{34, 1597, 28657},
{6765},
{144, 610},
{17711, 196418, 514229},
{2584, 4181, 10946, 121393, 317811},
{3524578}, {46368, 1346269, 1836311903},
{75025, 5702887, 24157817, 102334155, 165580141},
{832040, 14930352, 701408733}

T. D. Noe, Rows n = 0..500 of irregular triangle, flattened

Cf. A004685 (Fibonacci numbers in binary), A214853 (one 0-bit), A222601.

Cf. A222295, A222296, A222757, A222758.

f = Fibonacci[Range[0,1000]]; Table[Select[f, Count[IntegerDigits[#, 2], 0] == n &], {n, 0, 20}]

A222295 Conjectured number of Fibonacci numbers with exactly n bits set in their binary representation.

Original entry on oeis.org

1, 4, 4, 2, 3, 3, 4, 1, 4, 2, 1, 5, 5, 2, 2, 2, 5, 4, 3, 2, 2, 2, 3, 5, 3, 3, 2, 4, 2, 1, 4, 3, 2, 3, 3, 1, 6, 3, 2, 3, 3, 4, 4, 5, 0, 0, 3, 3, 2, 2, 5, 4, 3, 1, 5, 2, 2, 2, 5, 7, 3, 0, 0, 1, 2, 7, 3, 3, 2, 4, 3, 1, 2, 4, 4, 2, 0, 3, 1, 3, 7, 3, 4, 1, 3, 4, 3

Offset: 0

T. D. Noe, Feb 22 2013

			We set a(1) = 4 because Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, and Fib(6) = 8.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004685 (Fibonacci numbers in binary), A221158 (two bits set), A222296.
Cf. A011373 (number of bits set in each Fibonacci number).
Cf. A222601, A222602, A222757, A222758.

f = Fibonacci[Range[0,500]]; Table[Length[Select[f, Total[IntegerDigits[#, 2]] == n &]], {n, 0, 87}]

A221158 Fibonacci numbers with two 1's in the binary representation.

Original entry on oeis.org

3, 5, 34, 144

Offset: 1

Alex Ratushnyak, Feb 20 2013

Fibonacci numbers of the form 2^a + 2^b, a>b.
Elkies (2014) proved that there are no other terms.
This sequence is one row of A222296. - T. D. Noe, Mar 08 2013

			144 = 128 + 16 = 2^7 + 2^4, thus it is in the sequence.

Noam D. Elkies, Fibonacci numbers with Hamming weight 2, Mathoverflow, 2014.

Cf. A000045, A004685, A222296.

Select[Fibonacci[Range[1000]], DigitCount[#, 2, 1] == 2 &] (* Alonso del Arte, Feb 21 2013 *)

from sympy import fibonacci
print([f for n in range(100) if (f := int(fibonacci(n))).bit_count() == 2]) # David Radcliffe, Jul 03 2025

full, fini keywords added by Max Alekseyev, May 13 2014

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A222601 Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.

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A222602 Irregular triangle of conjectured Fibonacci numbers with exactly n 0-bits in their binary representation.

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A222295 Conjectured number of Fibonacci numbers with exactly n bits set in their binary representation.

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A221158 Fibonacci numbers with two 1's in the binary representation.

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