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a(n) = floor(n*tau) + n + 1 where tau is the golden ratio A001622.
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%I #60 Aug 05 2023 21:48:56

%S 1,3,6,8,11,14,16,19,21,24,27,29,32,35,37,40,42,45,48,50,53,55,58,61,

%T 63,66,69,71,74,76,79,82,84,87,90,92,95,97,100,103,105,108,110,113,

%U 116,118,121,124,126,129,131,134,137,139,142,144

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

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

%C 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

%C 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]

%C 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

%C a(n) = position of n-th 0 in A096270. - _Clark Kimberling_, Apr 22 2011

%C 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.)

%H Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, <a href="http://library.msri.org/books/Book70/files/1015.pdf">Geometric analysis of a generalized Wythoff game</a>, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.

%H U. Larsson and N. Fox, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Larsson/larsson8.html">An Aperiodic Subtraction Game of Nim-Dimension Two</a>, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

%H Ali Sada, <a href="http://list.seqfan.eu/pipermail/seqfan/2023-June/074685.html">Should A000201 and A026352 be cross-referenced?</a>, Seqfan thread, Jun 2023.

%t Table[Floor[GoldenRatio*n]+n+1,{n,0,60}] (* _Harvey P. Dale_, Aug 24 2021 *)

%o (PARI) a(n) = floor(n*(sqrt(5)+1)/2) + n + 1; \\ _Michel Marcus_, Sep 15 2016

%o (Python)

%o from math import isqrt

%o def A026352(n): return (n+isqrt(5*n**2)>>1)+n+1 # _Chai Wah Wu_, Aug 25 2022

%Y Essentially same as A004957.

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

%Y Subsequence of A000201.

%Y Complement of A026351.

%K nonn

%O 0,2

%A _Clark Kimberling_, Dec 11 1999