A102518 a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=0..k} binomial(3k, 3j).

1, 3, 27, 243, 2187, 19683, 177147, 1594323, 14348907, 129140163, 1162261467, 10460353203, 94143178827, 847288609443, 7625597484987, 68630377364883, 617673396283947, 5559060566555523, 50031545098999707, 450283905890997363, 4052555153018976267, 36472996377170786403

Offset: 0

Author

Paul Barry, Jan 13 2005

Keywords

• easy
• nonn

Comments

Binomial transform of A007613.

a(n+1) is the smallest number with a reciprocal with repeating decimal of period a(n). - Matthew Goers, Nov 09 2017

a(n) is the number of walks of 2n steps on the utility graph that start and end at the same vertex (excursions). A001019(n) is the number of 2n+1-step walks on the utility graph that end at one of the 3 adjacent vertices. A013708(n) is the number of 2n+2-step walks that end at one of the 2 remote vertices (at distance 2). The number of n-step walks on the utility (3-regular) graph, summed over all 3 types of final vertices, is 3^n. - R. J. Mathar, Nov 03 2020

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• R. J. Mathar, Counting Walks on Finite Graphs, Section 4.
• Index entries for linear recurrences with constant coefficients, signature (9).

Crossrefs

Cf. A001019, A007613, A013708.

Programs

Mathematica

Join[{1},NestList[9#&,3,20]] (* Harvey P. Dale, Feb 03 2021 *)

PARI

Vec((1-6*x)/(1-9*x) + O(x^30)) \\ Colin Barker, Mar 17 2016

Formula

a(n) = 3^(2*n-1) + 2*0^k/3; a(n+1) = A013708(n).

G.f.: (1-6*x) / (1-9*x). - Colin Barker, Mar 17 2016

E.g.f.: (exp(9*x) + 2)/3. - Stefano Spezia, Jul 09 2024