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

1,2

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.

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)

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}]

Crossrefs

Cf. A130600, A055778, A133775, A042948.

Sequence in context: A071570 A336047 A082448 * A070783 A198334 A055593

Adjacent sequences: A190793 A190794 A190795 * A190797 A190798 A190799

Keyword

nonn,base

Author

T. D. Noe, May 20 2011

Status

approved