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A102518
a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=0..k} binomial(3k, 3j).
19
1, 3, 27, 243, 2187, 19683, 177147, 1594323, 14348907, 129140163, 1162261467, 10460353203, 94143178827, 847288609443, 7625597484987, 68630377364883, 617673396283947, 5559060566555523, 50031545098999707, 450283905890997363, 4052555153018976267, 36472996377170786403
OFFSET
0,2
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
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
MATHEMATICA
Join[{1}, NestList[9#&, 3, 20]] (* Harvey P. Dale, Feb 03 2021 *)
PROG
(PARI) Vec((1-6*x)/(1-9*x) + O(x^30)) \\ Colin Barker, Mar 17 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 13 2005
STATUS
approved