A089910 Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n-1) = A005614(n-2) = 1.

4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237, 242, 245

Offset

1,1

Comments

a(n) is the number k such that floor(k/r) = floor(n*r^2), where r = golden ratio. - Clark Kimberling, May 03 2015

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

N. Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 2014.

U. Larsson and N. Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

Formula

a(n) = floor((2+sqrt(5))*n) + 0 or 1;

floor(n*(2+sqrt(5))) + b(a(n)) - a(n) = 0 where b(x) = A078588(x) = x + 1 + ceiling(x*sqrt(5)) - 2*ceiling(x*(1+sqrt(5))/2).

For n >= 2, a(n) = a(n-1) + d, where d = 5 if n-1 is in A000201, else d = 3. - Clark Kimberling, May 03 2015

a(n) = A003623(n) + 1 = A(B(n)) + 1, where A(B(n)) are the Wythoff AB-numbers. - Michel Dekking, Sep 15 2016

Maple

phi:=(1+sqrt(5))/2:  seq(floor(phi*floor(n*phi^2))+1, n=1..80); # Michel Dekking, Sep 15 2016

Mathematica

r = GoldenRatio; u = Flatten[Table[Select[Range[Floor[(r^2 + r) n], Floor[(r^2 + r) n + 1]], Floor[#/r] == Floor[n*r^2] &], {n, 1, 100}]] (* Clark Kimberling, May 03 2015 *)

Prog

(Python)
from math import isqrt
def A089910(n): return (n+isqrt(5*n**2)&-2)+n+1 # Chai Wah Wu, Aug 29 2022

Crossrefs

Cf. A000201, A001950, A026352, A270788.

Sequence in context: A312861 A301688 A276885 * A344275 A312862 A177880

Adjacent sequences: A089907 A089908 A089909 * A089911 A089912 A089913

Keyword

nonn

Author

Benoit Cloitre, Nov 15 2003

Extensions

Definition corrected by Jeffrey Shallit, Dec 21 2023

Status

approved