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
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathan Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
Urban Larsson and Nathan 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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Nov 15 2003
EXTENSIONS
Definition corrected by Jeffrey Shallit, Dec 21 2023
STATUS
approved