A194724 Number of quaternary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

1, 1, 7, 58, 523, 4966, 48838, 492724, 5068915, 52955950, 560198962, 5987822380, 64563867454, 701383563388, 7668869344108, 84326618668648, 931894610845123, 10344218506421758, 115280448164645818, 1289346114476360188, 14467472108268263818, 162816535672067515828

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013

Formula

G.f.: 3/4 + 3/(2*(2+4*sqrt(1-12*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*3^j for n>0.

a(n) ~ 3 * 12^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

Conjecture: n*a(n) +2*(-14*n+9)*a(n-1) +96*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 14 2015

Example

a(2) = 7: aaaa, aabb, aacc, aadd, abba, acca, adda (with quaternary alphabet {a,b,c,d}).

Maple

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *3^j, j=0..n-1)/n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 7][n+1],
      ((28*n-18)*a(n-1) -(192*n-288)*a(n-2))/n)
    end:
seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[3/4+3/(2(2+4Sqrt[1-12x])), {x, 0, 30}], x] (* Harvey P. Dale, Sep 30 2012 *)

Crossrefs

Column k=4 of A183134.

Sequence in context: A244469 A006193 A139397 * A308650 A081343 A163048

Adjacent sequences: A194721 A194722 A194723 * A194725 A194726 A194727

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved