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.

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

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

The first differences are A034891, the second differences are A000607.

Sequence in context: A039899 A039901 A173564 * A241823 A058984 A084376

Adjacent sequences: A121943 A121944 A121945 * A121947 A121948 A121949

Keyword

nonn,more,changed

Author

David S. Newman, Sep 04 2006

Status

approved