A331124 Number of function evaluations in a recursive calculation of Fibonacci(n).

1, 1, 1, 3, 6, 5, 11, 10, 15, 12, 23, 17, 27, 22, 37, 26, 38, 28, 51, 36, 53, 41, 68, 45, 67, 50, 87, 60, 86, 64, 102, 65, 93, 67, 118, 80, 116, 88, 141, 90, 131, 95, 163, 110, 155, 114, 181, 113, 163, 118, 205, 138, 198, 148, 234, 147, 211, 151, 253, 167

Offset

0,4

Comments

One way to calculate the Fibonacci numbers recursively is to use:

- F(0) = 0,

- F(1) = 1,

- F(2) = 1,

- F(n) = F((n + 1)/2)^2 + F((n - 1)/2)^2 for odd n,

- F(n) = F(n/2) * (F(n/2 - 1) + F(n/2 + 1)) for even n.

Proof: it is known that F(i) * F(j) + F(i + 1) * F(j + 1) = F(i + j + 1) (see formula section of A000045):

- for even n, let i = n/2 and j = n/2 - 1,

- for odd n, let i = j = (n + 1)/2.

This sequence gives the number of evaluations of F for calculating F(n). It is assumed that F needs to be evaluated more than once even if it has been evaluated before (no caching).

Conjecture: for large n, this sequence is bounded by a small constant times n^(4/3).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

GeeksforGeeks, Program for Fibonacci numbers, see method 6, but erroneously states to take O(log n) operations.

Rémy Sigrist, PARI program for A331124

Formula

a(0) = 1,

a(1) = 1,

a(2) = 1,

a(n) = a((n + 1)/2) + a((n - 1)/2) + 1, n odd and n > 2,

a(n) = a(n/2) + a(n/2 - 1) + a(n/2 + 1) + 1, n even.

Example

a(5) = a(3) + a(2) + 1 = (a(2) + a(1) + 1) + a(2) + 1 = 5.

Prog

(Fortran)
program main
  implicit none
  integer, parameter :: pmax = 100
  integer :: r, n
  logical, dimension(0:pmax) :: cache
  do n = 0, pmax
     cache (0:2) = .true.
     cache (3:pmax) = .false.
     r = a(n)
     write (*, fmt="(I0, ', ')", advance="no") r
  end do
  write (*, fmt="()")
contains
  recursive function a (n) result(r)
    integer, intent(in) :: n
    integer :: r
    if (cache(n)) then
       r = 1
       return
    else if (mod(n, 2) == 1) then
       r = a ((n + 1)/2) + a ((n - 1)/2) + 1
    else
       r = a (n/2) + a(n/2 - 1) + a(n/2 + 1) + 1
    end if
    cache (n) = .false.
  end function a
end program main
(PARI) \\ See Links section.

Crossrefs

Cf. A000045; see A331164 for the number of function evaluations with caching.

Sequence in context: A093419 A160049 A096620 * A299209 A007479 A076535

Adjacent sequences: A331121 A331122 A331123 * A331125 A331126 A331127

Keyword

nonn,easy,look

Author

Thomas König, Jan 10 2020

Status

approved