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A119346 Sequence of nim-values for the game in which two players alternately cut off one inch or root two inches from a piece of string of length n. Player who runs out of string loses. 1
0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Michel Dekking, Feb 17 2020: (Start)

It follows from Alex Fink's remarks that (a(n)) is obtained from the sequence A276862 (removing the first 2) by mapping every 2 to 0,1 and every 3 to 0,1,2. However, the first 3 entries will be missing.

In the context of my paper "Morphic words, Beatty sequences and integer images of the Fibonacci language", this means that (a(n+3)) is obtained by decorating A006337 by the decoration delta given by delta(1) = 01, delta(2) = 012. This implies that (a(n+3)) is a morphic sequence, i.e., the letter to letter image of the fixed point of a morphism, say sigma. One obtains sigma by the 'natural' algorithm given in the "Morphic words...."-paper. In turns out that the alphabet of sigma can be chosen as {0,1,2}, and that sigma is surprisingly simple:

      sigma(0) = 01, sigma(1) = 012, sigma(2) = 01.

The letter to letter map is given by the identity. In other words, if x = 010120101... is the unique fixed point of sigma, then (a(n+3)) = x. (End)

LINKS

Table of n, a(n) for n=0..98.

M. Dekking, Morphic words, Beatty sequences and integer images of the Fibonacci language, Theoretical Computer Science  809,  407-417 (2020).

Alex Fink, Discussion of A119346

FORMULA

To get the answers, add one to sequence A003151 and then start counting from zero, but return to zero whenever you reach a member of A003151 plus one.

Added Feb 13 2020: The simplest formula is a(n) = floor(n mod (1 + sqrt 2)). - Alex Fink (see link).

CROSSREFS

Cf. A003151.

Sequence in context: A277731 A298307 A287002 * A014586 A122924 A133450

Adjacent sequences:  A119343 A119344 A119345 * A119347 A119348 A119349

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, based on email from R. K. Guy and Alex Fink, Aug 05 2006

STATUS

approved

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Last modified June 6 13:31 EDT 2020. Contains 334827 sequences. (Running on oeis4.)