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%I #43 May 14 2024 17:20:34
%S 1,7,34,142,547,2005,7108,24604,83653,280483,930022,3055786,9964519,
%T 32285041,104029576,333612088,1065406345,3389929279,10750918570,
%U 33996147910,107218620331,337346390797,1059110761804,3318547053652,10379285465677,32408789311195
%N a(1)=1, a(n) = n*3^(n-1) + a(n-1).
%H Vincenzo Librandi, <a href="/A014915/b014915.txt">Table of n, a(n) for n = 1..600</a>
%H A. V. Kitaev, <a href="https://arxiv.org/abs/1809.00122">Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin</a>, arXiv:1809.00122 [math.CA], 2018.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,9).
%F From _Henry Bottomley_, Dec 18 2000: (Start)
%F a(n) = ((2*n-1)*3^n + 1)/4;
%F a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3);
%F a(n) = 1 + 2*3 + 3*3^2 + .. + n*3^(n-1);
%F a(n) = a(n-1) + A027471(n+1). (End)
%F G.f.: x/((1-x)*(1-3*x)^2). - _Colin Barker_, Jul 28 2012
%F a(n) = f^n(n)/2 with f(x) = 3*x-1. - _Glen Gilchrist_, Apr 10 2019
%F E.g.f.: exp(x)*(1 + exp(2*x)*(6*x - 1))/4. - _Stefano Spezia_, May 14 2024
%t LinearRecurrence[{7, -15, 9}, {1, 7, 34}, 25] (* _L. Edson Jeffery_, May 08 2015 *)
%o (Magma) [((2*n - 1)*3^n + 1)/4: n in [1..30]]; // _Vincenzo Librandi_, Jun 09 2011
%Y Cf. A027471, A059045.
%Y Cf. A027261, A064017, A079272.
%K nonn,easy
%O 1,2
%A _Olivier Gérard_
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