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A121946 Let r be the matrix {{1,1},{0,1}} and b={{1,0},{1,0}}. Let A be the semigroup generated by r and b. a(n) is the number of words of length n in A. 0
1, 2, 3, 5, 8, 12, 18, 26, 37, 51, 69, 92, 121, 157, 202, 257, 324, 405, 503, 620, 760, 926, 1122, 1353, 1624, 1941, 2310, 2739, 3235, 3808, 4468, 5226, 6095, 7088, 8221, 9511, 10976, 12638, 14519, 16644, 19041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The generating function was found by Moshe Shmuel Newman.

LINKS

Table of n, a(n) for n=0..40.

Melvyn B. Nathanson, Number Theory and semigroups of intermediate growth, Amer. Math. Monthly 106(1999) 666-669.

FORMULA

G.f.: 1/(1-x)+ x/((1-x)^2 (1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)...) where the product is over all primes.

EXAMPLE

The products of two matrices in A are r.r, r.b, b.r and b.b, that is {{1, 2}, {0, 1}}, {{2, 0}, {1, 0}}, {{1, 1}, {1, 1}}, {{1, 0}, {1, 0}}.

Of these, the last one, b.b is an element of length one, since it is equal to b. The remainder are elements of length two, hence a(2)=3.

CROSSREFS

Sequence in context: A039899 A039901 A173564 * A241823 A058984 A084376

Adjacent sequences:  A121943 A121944 A121945 * A121947 A121948 A121949

KEYWORD

nonn

AUTHOR

David S. Newman, Sep 04 2006

STATUS

approved

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Last modified May 11 19:29 EDT 2021. Contains 343808 sequences. (Running on oeis4.)