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a(n) = (7*8^n + 2*(-1)^n)/9.
2

%I #18 Sep 06 2024 14:30:14

%S 1,6,50,398,3186,25486,203890,1631118,13048946,104391566,835132530,

%T 6681060238,53448481906,427587855246,3420702841970,27365622735758,

%U 218924981886066,1751399855088526,14011198840708210,112089590725665678,896716725805325426,7173733806442603406

%N a(n) = (7*8^n + 2*(-1)^n)/9.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,8).

%F a(n) = Sum_{j=0..2} Sum_{k=0..n} C(3*n+j, 3*k)/3.

%F a(n) = (A007613(n) + A082311(n) + A082365(n))/3.

%F G.f.: (-1+x)/((1+x)*(8*x-1)). - _R. J. Mathar_, Dec 10 2014

%F From _Elmo R. Oliveira_, Aug 18 2024: (Start)

%F E.g.f.: exp(-x)*(7*exp(9*x) + 2)/9.

%F a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

%t LinearRecurrence[{7,8},{1,6},20] (* _Harvey P. Dale_, Aug 15 2016 *)

%Y First differences of A015565.

%Y Cf. A007613, A082311, A082365.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Nov 29 2003

%E a(20)-a(21) from _Elmo R. Oliveira_, Aug 18 2024