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A014896 a(1) = 1, a(n) = 13*a(n-1) + n.

%I A014896 #42 Mar 31 2025 13:38:39
%S A014896 1,15,198,2578,33519,435753,5664796,73642356,957350637,12445558291,
%T A014896 161792257794,2103299351334,27342891567355,355457590375629,
%U A014896 4620948674883192,60072332773481512,780940326055259673,10152224238718375767,131978915103338884990,1715725896343405504890
%N A014896 a(1) = 1, a(n) = 13*a(n-1) + n.
%H A014896 Vincenzo Librandi, <a href="/A014896/b014896.txt">Table of n, a(n) for n = 1..200</a>
%H A014896 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-27,13).
%F A014896 a(n) = 15*a(n-1) - 27*a(n-2) + 13*a(n-3), with a(1)=1, a(2)=15, a(3)=198. - _Vincenzo Librandi_, Oct 20 2012
%F A014896 G.f.: x/((1-13*x)*(1-x)^2). - _Jinyuan Wang_, Mar 11 2020
%F A014896 From _Elmo R. Oliveira_, Mar 31 2025: (Start)
%F A014896 E.g.f.: exp(x)*(13*exp(12*x) - 12*x - 13)/144.
%F A014896 a(n) = (13^(n+1) - 12*n - 11)/144. (End)
%p A014896 a:=n->sum((13^(n-j)-1)/12,j=0..n): seq(a(n), n=1..17); # _Zerinvary Lajos_, Jan 05 2007
%p A014896 a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,13]])^n)[2,3]:
%p A014896 seq(a(n), n=1..17);  # _Alois P. Heinz_, Aug 06 2008
%t A014896 LinearRecurrence[{15, -27, 13}, {1, 15, 198}, 20] (* _Vincenzo Librandi_, Oct 20 2012 *)
%o A014896 (Magma) I:=[1, 15, 198]; [n le 3 select I[n] else 15*Self(n-1) - 27*Self(n-2)+ 13*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Oct 20 2012
%o A014896 (Maxima)
%o A014896 a[1]:1$
%o A014896 a[2]:15$
%o A014896 a[3]:198$
%o A014896 a[n]:=15*a[n-1]-27*a[n-2]+13*a[n-3]$
%o A014896 A014896(n):=a[n]$ makelist(A014896(n),n,1,30); /* _Martin Ettl_, Nov 07 2012 */
%Y A014896 Row n=13 of A126885.
%K A014896 nonn,easy
%O A014896 1,2
%A A014896 _N. J. A. Sloane_, _Olivier Gérard_