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A353097 a(1) = 5; for n > 1, a(n) = 6*a(n-1) + 6 - n.

%I A353097 #27 May 28 2023 21:52:31
%S A353097 5,34,207,1244,7465,44790,268739,1612432,9674589,58047530,348285175,
%T A353097 2089711044,12538266257,75229597534,451377585195,2708265511160,
%U A353097 16249593066949,97497558401682,584985350410079,3509912102460460,21059472614762745,126356835688576454
%N A353097 a(1) = 5; for n > 1, a(n) = 6*a(n-1) + 6 - n.
%H A353097 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-13,6).
%F A353097 G.f.: x * (5 - 6*x)/((1 - x)^2 * (1 - 6*x)).
%F A353097 a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3).
%F A353097 a(n) = 4*A014829(n) + n.
%F A353097 a(n) = (4*6^(n+1) + 5*n - 24)/25.
%F A353097 a(n) = Sum_{k=0..n-1} (6 - n + k) * 6^k.
%F A353097 E.g.f.: exp(x)*(24*exp(5*x) + 5*x - 24)/25. - _Stefano Spezia_, May 28 2023
%t A353097 LinearRecurrence[{8, -13, 6}, {5, 34, 207}, 22] (* _Amiram Eldar_, Apr 23 2022 *)
%o A353097 (PARI) my(N=30, x='x+O('x^N)); Vec(x*(5-6*x)/((1-x)^2*(1-6*x)))
%o A353097 (PARI) a(n) = (4*6^(n+1)+5*n-24)/25;
%o A353097 (PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
%o A353097 a(n) = b(n, 6);
%Y A353097 Cf. A064617, A353094, A353095, A353096, A353098, A353099, A353100.
%Y A353097 Cf. A014829.
%K A353097 nonn,easy
%O A353097 1,1
%A A353097 _Seiichi Manyama_, Apr 23 2022