

A089910


Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n) = A005614(n+1) = 1.


6



4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237, 242, 245
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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
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
N. Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
U. Larsson, N. Fox, An Aperiodic Subtraction Game of NimDimension 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(n1) + d, where d = 5 if n1 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 ABnumbers.  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 *)


CROSSREFS

Cf. A000201, A001950, A026352, A270788.
Sequence in context: A312861 A301688 A276885 * A312862 A177880 A059269
Adjacent sequences: A089907 A089908 A089909 * A089911 A089912 A089913


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Nov 15 2003


STATUS

approved



