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%I #51 Dec 21 2023 15:15:35
%S 4,9,12,17,22,25,30,33,38,43,46,51,56,59,64,67,72,77,80,85,88,93,98,
%T 101,106,111,114,119,122,127,132,135,140,145,148,153,156,161,166,169,
%U 174,177,182,187,190,195,200,203,208,211,216,221,224,229,232,237,242,245
%N Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n-1) = A005614(n-2) = 1.
%C a(n) is the number k such that floor(k/r) = floor(n*r^2), where r = golden ratio. - _Clark Kimberling_, May 03 2015
%H Clark Kimberling, <a href="/A089910/b089910.txt">Table of n, a(n) for n = 1..1000</a>
%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H N. Fox, <a href="http://arxiv.org/abs/1407.2823">On Aperiodic Subtraction Games with Bounded Nim Sequence</a>, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
%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.
%F a(n) = floor((2+sqrt(5))*n) + 0 or 1;
%F 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).
%F 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
%F a(n) = A003623(n) + 1 = A(B(n)) + 1, where A(B(n)) are the Wythoff AB-numbers. - _Michel Dekking_, Sep 15 2016
%p phi:=(1+sqrt(5))/2: seq(floor(phi*floor(n*phi^2))+1, n=1..80); # _Michel Dekking_, Sep 15 2016
%t 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 *)
%o (Python)
%o from math import isqrt
%o def A089910(n): return (n+isqrt(5*n**2)&-2)+n+1 # _Chai Wah Wu_, Aug 29 2022
%Y Cf. A000201, A001950, A026352, A270788.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Nov 15 2003
%E Definition corrected by _Jeffrey Shallit_, Dec 21 2023
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