cp's OEIS Frontend

A026352 a(n) = floor(n*tau) + n + 1 where tau is the golden ratio A001622.

%I A026352 #71 Jul 28 2025 10:54:23
%S A026352 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 A026352 63,66,69,71,74,76,79,82,84,87,90,92,95,97,100,103,105,108,110,113,
%U A026352 116,118,121,124,126,129,131,134,137,139,142,144
%N A026352 a(n) = floor(n*tau) + n + 1 where tau is the golden ratio A001622.
%C A026352 a(n) = greatest k such that s(k) = n+1, where s = A026350.
%C A026352 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 A026352 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 A026352 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 A026352 a(n) = position of n-th 0 in A096270. - _Clark Kimberling_, Apr 22 2011
%C A026352 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.)
%C A026352 In the Fokkink-Joshi paper, this sequence is the Cloitre (1,1,2,3)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n-1, else a(n) = a(n-1)+3. - _Michael De Vlieger_, Jul 28 2025
%H A026352 Benoit Cloitre, <a href="https://arxiv.org/abs/2506.18103">A study of a family of self-referential sequences</a>, arXiv:2506.18103 [math.GM], 2025. See p. 9.
%H A026352 Robbert Fokkink and Gandhar Joshi, <a href="https://arxiv.org/abs/2507.16956">On Cloitre's hiccup sequences</a>, arXiv:2507.16956 [math.CO], 2025. See pp. 8, 17.
%H A026352 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 A026352 Urban Larsson and Nathan 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 A026352 Ali Sada, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/pipermail/seqfan/2023-June/074685.html">Should A000201 and A026352 be cross-referenced?</a>, Seqfan thread, Jun 2023.
%F A026352 a(n) = A026351(n)+n. - _R. J. Mathar_, Jun 24 2025
%t A026352 Table[Floor[GoldenRatio*n]+n+1,{n,0,60}] (* _Harvey P. Dale_, Aug 24 2021 *)
%t A026352 cloitreH[j_, x_, y_, z_, w_ : 120] := Block[{c, k}, c[_] := False; k = x; c[x] = True; {x}~Join~Reap[Do[If[c[n - j], k += y, k += z]; c[k] = True; Sow[k], {n, 2, w}] ][[-1, 1]] ]; cloitreH[1, 1, 2, 3] (* _Michael De Vlieger_, Jul 28 2025 *)
%o A026352 (PARI) a(n) = floor(n*(sqrt(5)+1)/2) + n + 1; \\ _Michel Marcus_, Sep 15 2016
%o A026352 (Python)
%o A026352 from math import isqrt
%o A026352 def A026352(n): return (n+isqrt(5*n**2)>>1)+n+1 # _Chai Wah Wu_, Aug 25 2022
%Y A026352 Essentially same as A004957.
%Y A026352 Cf. A001622, A005614, A026350, A095791, A096270.
%Y A026352 Subsequence of A000201.
%Y A026352 Complement of A026351.
%K A026352 nonn
%O A026352 0,2
%A A026352 _Clark Kimberling_, Dec 11 1999