A190796 Number of digits in the minimal base-phi representation of n.

1, 4, 5, 5, 8, 8, 9, 9, 9, 9, 9, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

Author

T. D. Noe, May 20 2011

Keywords

• nonn
• base

Comments

See A130600(n) for the digits in the minimal base phi representation of n.

a(n) <= 2 * ceiling( log(n) / log(phi) ) for n > 1.

References

• Michel Dekking and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." Fib. Quart. 61:2 (May 2023), 105-118.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.

Crossrefs

Cf. A130600, A055778, A133775, A042948.

Programs

Mathematica

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; last1, {n, 1, nn}]

Formula

a(n) = A055778(n) + A133775(n).

From Michel Dekking, Jun 19 2024: (Start)

Let (L(n)) = (2, 1, 3, 4, 7, 11, 18, 29, 47, ...) = A000032 be the Lucas numbers.

If L(2n) <= i <= L(2n+1), then a(i) = 4n+1; if L(2n+1)+1 <= i < L(2n+2), then a(i) = 4n+4.

This formula follows from Proposition 4.2. in "On the representation of the natural numbers by powers of the golden mean".

For example if n=1: L(2)=3, L(3)=4, L(4)=7, so a(3) = a(4) = 5, and a(5) = a(6) = 8.

Let (v(n)) = 1,4,5,8,9,12,... be the sequence of values taken by (a(n)). Then it follows directly from the Lucas formula for (a(n)) that v(n) = A042948(n) (where A042948 has been given offset 1, as it should; see also the comment by Jianing Song in A042948).

(End)