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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 (list; graph; refs; listen; history; text; internal format)
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
LINKS
R. J. Mathar, Counting Walks on Finite Graphs, Section 4.
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
Sequence in context: A344724 A268094 A013708 * A361842 A168495 A037763
KEYWORD
easy,nonn,changed
AUTHOR
Paul Barry, Jan 13 2005
STATUS
approved

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Last modified July 13 08:08 EDT 2024. Contains 374274 sequences. (Running on oeis4.)