A352964 a(0) = 0, a(1) = 1, and for any n > 1, a(n) = a(n-F(e)) + a(n-F(e+1)) with e as large as possible (where F(k) is the k-th Fibonacci number).

0, 1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 3, 5, 1, 3, 4, 2, 5, 3, 5, 8, 1, 4, 3, 4, 7, 2, 5, 7, 3, 8, 5, 8, 13, 1, 4, 5, 3, 7, 4, 7, 11, 2, 7, 5, 7, 12, 3, 8, 11, 5, 13, 8, 13, 21, 1, 5, 4, 5, 9, 3, 7, 10, 4, 11, 7, 11, 18, 2, 7, 9, 5, 12, 7, 12, 19, 3, 11, 8, 11, 19

Offset

0,5

Comments

This sequence is a variant of the Fibonacci sequence (A000045) with variable steps.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10945

Rémy Sigrist, Logarithmic scatterplot of the first F(27) = 196418 terms

Formula

a(A000045(k)) = 1 for any k > 0.

Example

a(0) = 0 by definition.
a(1) = 1 by definition.
a(2) = a(2-F(2)) + a(2-F(3)) = a(1) + a(0) = 1 + 0 = 1.
a(3) = a(3-F(3)) + a(3-F(4)) = a(1) + a(0) = 1 + 0 = 1.
a(4) = a(4-F(3)) + a(4-F(4)) = a(2) + a(1) = 1 + 1 = 2.
a(5) = a(5-F(4)) + a(5-F(5)) = a(2) + a(0) = 1 + 0 = 1.
a(6) = a(6-F(4)) + a(6-F(5)) = a(3) + a(1) = 1 + 1 = 2.
a(7) = a(7-F(4)) + a(7-F(5)) = a(4) + a(2) = 2 + 1 = 3.
a(8) = a(8-F(5)) + a(8-F(6)) = a(3) + a(0) = 1 + 0 = 1.

Prog

(PARI) { e=2; for (n=1, #a=vector(81), print1 (a[n]=if (n==1, 0, n==2, 1, if (n>fibonacci(e+2), e++); a[n-fibonacci(e)]+a[n-fibonacci(e+1)]), ", ")) }

Crossrefs

See A352961 for a similar sequence.

Cf. A000045, A005185, A072649.

Sequence in context: A048793 A344084 A249783 * A209278 A326921 A370408

Adjacent sequences: A352961 A352962 A352963 * A352965 A352966 A352967

Keyword

nonn

Author

Rémy Sigrist, Apr 11 2022

Status

approved