A289749 Number of ways not ending in 011 to write n in base phi.

1, 1, 2, 3, 3, 5, 5, 5, 8, 8, 8, 5, 10, 13, 12, 12, 13, 10, 7, 15, 18, 21, 16, 20, 20, 16, 21, 18, 15, 7, 17, 25, 27, 27, 34, 29, 20, 32, 32, 32, 20, 29, 34, 27, 27, 25, 17, 9, 24, 32, 40, 33, 45, 45, 39, 55, 50, 45, 24, 40, 52, 48, 48, 52, 40, 24, 45, 50, 55, 39, 45, 45

Offset: 0

Author

Gilian Breysens, Jul 11 2017

Keywords

• nonn
• base

Comments

Old name was: Number of ways to write n in base phi.

phi = (1+sqrt(5))/2. Base phi is also called golden ratio base or phinary. In base phi, we can replace two consecutive 1's with a one in the column to the left; e.g., "011" = "100".

Conjecture: a(A005248(k)) = 2k+1 for k=1,2,...(cf. Theorem 2 in the paper by Carlitz.) - Michel Dekking, Nov 14 2021

This conjecture is proved in the paper "Counting base phi representations". - Michel Dekking, Jul 15 2023

Examples

			a(3) = 3, because 3 in base phi = 10.1111 = 11.01 = 100.01.
		

Links

• L. Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220.
• Michel Dekking and Ad van Loon, Counting base phi representations, arXiv:2304.11387 [math.NT], 2023.
• Ron Knott, Base phi calculator.

Crossrefs

Cf. A001622, A130600, A130601.

Extensions

Name corrected by Michel Dekking, Sep 09 2021