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

%I A102518 #43 Jul 09 2024 12:45:10
%S A102518 1,3,27,243,2187,19683,177147,1594323,14348907,129140163,1162261467,
%T A102518 10460353203,94143178827,847288609443,7625597484987,68630377364883,
%U A102518 617673396283947,5559060566555523,50031545098999707,450283905890997363,4052555153018976267,36472996377170786403
%N A102518 a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=0..k} binomial(3k, 3j).
%C A102518 Binomial transform of A007613.
%C A102518 a(n+1) is the smallest number with a reciprocal with repeating decimal of period a(n). - _Matthew Goers_, Nov 09 2017
%C A102518 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
%H A102518 Colin Barker, <a href="/A102518/b102518.txt">Table of n, a(n) for n = 0..1000</a>
%H A102518 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Section 4.
%H A102518 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (9).
%F A102518 a(n) = 3^(2*n-1) + 2*0^k/3; a(n+1) = A013708(n).
%F A102518 G.f.: (1-6*x) / (1-9*x). - _Colin Barker_, Mar 17 2016
%F A102518 E.g.f.: (exp(9*x) + 2)/3. - _Stefano Spezia_, Jul 09 2024
%t A102518 Join[{1},NestList[9#&,3,20]] (* _Harvey P. Dale_, Feb 03 2021 *)
%o A102518 (PARI) Vec((1-6*x)/(1-9*x) + O(x^30)) \\ _Colin Barker_, Mar 17 2016
%Y A102518 Cf. A001019, A007613, A013708.
%K A102518 easy,nonn
%O A102518 0,2
%A A102518 _Paul Barry_, Jan 13 2005