A101168 Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> a}.

1, 3, 9, 25, 71, 201, 569, 1611, 4561, 12913, 36559, 103505, 293041, 829651, 2348889, 6650121, 18827671, 53304473, 150914409, 427265435, 1209664161, 3424773601, 9696140959, 27451493281, 77720042081, 220039211683, 622970000809, 1763738467065, 4993456147431

Offset

0,2

Links

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (2, 2, 1).

Formula

a(n) = 2*a(n-1) + 2*a(n-2) + a(n-3).

G.f.: (1+x+x^2) / (1-2*x-2*x^2-x^3).

a(n-1) = sum(k=1..n, sum(m=0..n-k, (sum(j=0..k, binomial(j, n-m-3*k+2*j) *binomial(k, j))) *sum(i=ceiling(m/2)..m, binomial(i, m-i)*binomial(k+i-1, k-1)))). - Vladimir Kruchinin, May 05 2011

Example

a => aab => aabaabaac => aabaabaacaabaabaacaabaaba, thus a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25.

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <1|2|2>>^n. <<1, 3, 9>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, May 06 2011

Mathematica

Length/@SubstitutionSystem[{a->{a, a, b}, b->{a, a, c}, c->a}, {a}, 15] (* The program generates the first 16 terms of the sequence. To generate more, increase the final ("15") constant. *) (* Harvey P. Dale, Sep 05 2022 *)

Prog

(Maxima) a(n):=b(n+1);
b(n):= sum(sum((sum(binomial(j, n+1-m-3*k+2*j) *binomial(k, j), j, 0, k)) *sum(binomial(i, m-i) *binomial(k+i-1, k-1), i, ceiling(m/2), m), m, 0, n+1-k), k, 1, n+1); /* Vladimir Kruchinin, May 05 2011 */

Crossrefs

Cf. A101399, A101400.

Sequence in context: A309105 A101197 A233828 * A211287 A211290 A211299

Adjacent sequences: A101165 A101166 A101167 * A101169 A101170 A101171

Keyword

easy,nonn

Author

Jeroen F.J. Laros, Jan 22 2005

Status

approved