A219660 - OEIS

A219660

a(n) = number of bit-positions where Fibonacci numbers F(n) and F(n+1) contain both an 1-bit in their binary representation.

0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 1, 4, 2, 3, 4, 3, 1, 4, 3, 1, 5, 4, 3, 3, 5, 7, 8, 4, 4, 3, 4, 8, 5, 4, 6, 6, 4, 7, 7, 10, 7, 11, 7, 8, 8, 4, 8, 12, 8, 9, 7, 8, 10, 13, 8, 8, 10, 8, 6, 12, 11, 12, 13, 10, 8, 7, 10, 13, 9, 9, 14, 12, 11, 9, 11, 13, 13, 13

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OFFSET

0,8

COMMENTS

This sequence gives the number of "first-level" carries produced when computing Fibonacci numbers in binary arithmetic. that is, the carry-1-bits produced at the positions where the both summands F(n) and F(n+1) have 1-bits in the same bit-positions. This sum doesn't include any additional carries produced, when a produced carry-bit is added to an existing 1 at its left side.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = A000120(A051122(n)).

EXAMPLE

F_7 = 13, ......01101 in binary.

F_8 = 21, ......10101 in binary.

--------------------------

Anded together: 00101

which has two 1-bits, thus a(7)=2.

MATHEMATICA

a[n_] := DigitCount[BitAnd[Fibonacci[n], Fibonacci[n+1]], 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 22 2023 *)