A307451 Sum of binary weights of two consecutive Fibonacci numbers minus the binary weight of the following Fibonacci number.

0, 1, 0, 1, 3, 0, 1, 4, 0, 3, 7, 1, 1, 7, 2, 5, 11, 6, 1, 7, 6, 3, 13, 11, 3, 4, 9, 10, 15, 9, 10, 7, 10, 14, 11, 9, 16, 12, 6, 11, 19, 15, 12, 19, 12, 23, 22, 7, 12, 19, 18, 17, 18, 11, 17, 27, 13, 20, 28, 17, 9, 22, 29, 18, 26, 18, 30, 18, 15, 24, 20, 20, 28, 28, 24, 24, 18, 21, 28

Offset

2,5

Comments

The binary weight of a positive Fibonacci number is at least 1 (and at least 2 for positive Fibonacci numbers other than 1, 2, 8) but not more than the sum of the binary weights of the previous two Fibonacci numbers.

Therefore a(n) is at least 0, at most n - 1.

Number of carries in base-2 addition of A000045(n-2)+A000045(n-1)=A000045(n). - Robert Israel, Apr 14 2019

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Dario Carrasquel, 5 ways to solve Fibonacci in Scala - Tail Recursion, Memoization, The Pisano Period & More, August 26, 2016.

Formula

a(n) = (A011373(n - 2) + A011373(n - 1)) - A011373(n).

Example

Fibonacci(8) = 21 = 10101 in binary.
Fibonacci(9) = 34 = 100010 in binary.
Fibonacci(10) = 55 = 110111 in binary, which has five 1s. We see that 10101 has three 1s and 100010 just two. Thus a(10) = 0.

Maple

B:= map(t -> convert(convert(combinat:-fibonacci(t), base, 2), `+`), [$0..100]):
B[1..-3]-B[2..-2]-B[3..-1]; # Robert Israel, Apr 14 2019

Mathematica

Table[(DigitCount[Fibonacci[n - 2], 2, 1] + DigitCount[Fibonacci[n - 1], 2, 1]) - DigitCount[Fibonacci[n], 2, 1], {n, 2, 100}]

Prog

(Scala) def fibonacci(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 0, 1)
} // Based on "Case 3: Tail Recursion" from Carrasquel (2016) link
(2 to 100).map(n => (fibonacci(n - 2).bitCount + fibonacci(n - 1).bitCount) - fibonacci(n).bitCount)
(PARI) f(n) = hammingweight(fibonacci(n)); \\ A011373
a(n) = f(n-1) + f(n-2) - f(n); \\ Michel Marcus, Apr 14 2019

Crossrefs

Cf. A000045, A000120, A011373.

Sequence in context: A346203 A327126 A273083 * A247629 A178116 A238709

Adjacent sequences: A307448 A307449 A307450 * A307452 A307453 A307454

Keyword

nonn,easy,base

Author

Alonso del Arte, Apr 08 2019

Status

approved