A026352 - OEIS

A026352

a(n) = floor(n*tau) + n + 1 where tau is the golden ratio A001622.

1, 3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, 35, 37, 40, 42, 45, 48, 50, 53, 55, 58, 61, 63, 66, 69, 71, 74, 76, 79, 82, 84, 87, 90, 92, 95, 97, 100, 103, 105, 108, 110, 113, 116, 118, 121, 124, 126, 129, 131, 134, 137, 139, 142, 144

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = greatest k such that s(k) = n+1, where s = A026350.

Indices at which blocks (0;1) occur in infinite Fibonacci word; i.e., n such that A005614(n)=0 and A005614(n+1)=1. - Benoit Cloitre, Nov 15 2003

Except for the first term, these are the numbers whose lazy Fibonacci representation (see A095791) includes both 1 and 2; thus this sequence is a subsequence of the lower Wythoff sequence, A000201. - Clark Kimberling, Jun 10 2004 [A-number typo corrected by Nathan Fox, May 03 2014]

a(n) = n-th number k whose lazy Fibonacci representation (as in A095791) has more summands than that of k-1. - Clark Kimberling, Jun 12 2004

a(n) = position of n-th 0 in A096270. - Clark Kimberling, Apr 22 2011

Maximum number of chips in a pile created at each step in the game described by Roland Schroeder in his comment at A000201. (From Allan C. Wechsler via Seqfan.)

LINKS

Table of n, a(n) for n=0..55.

Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.

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

Ali Sada, Should A000201 and A026352 be cross-referenced?, Seqfan thread, Jun 2023.

MATHEMATICA

Table[Floor[GoldenRatio*n]+n+1, {n, 0, 60}] (* Harvey P. Dale, Aug 24 2021 *)

PROG

(PARI) a(n) = floor(n*(sqrt(5)+1)/2) + n + 1; \\ Michel Marcus, Sep 15 2016

(Python)

from math import isqrt

def A026352(n): return (n+isqrt(5*n**2)>>1)+n+1 # Chai Wah Wu, Aug 25 2022

CROSSREFS

Essentially same as A004957.

Cf. A001622, A005614, A026350, A095791, A096270.