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A353096 a(1) = 4; for n > 1, a(n) = 5*a(n-1) + 5 - n.

%I A353096 #30 May 28 2023 22:21:21
%S A353096 4,23,117,586,2930,14649,73243,366212,1831056,9155275,45776369,
%T A353096 228881838,1144409182,5722045901,28610229495,143051147464,
%U A353096 715255737308,3576278686527,17881393432621,89406967163090,447034835815434,2235174179077153,11175870895385747
%N A353096 a(1) = 4; for n > 1, a(n) = 5*a(n-1) + 5 - n.
%H A353096 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11,5).
%F A353096 G.f.: x * (4 - 5*x)/((1 - x)^2 * (1 - 5*x)).
%F A353096 a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
%F A353096 a(n) = 3*A014827(n) + n.
%F A353096 a(n) = (3*5^(n+1) + 4*n - 15)/16.
%F A353096 a(n) = Sum_{k=0..n-1} (5 - n + k) * 5^k.
%F A353096 E.g.f.: exp(x)*(15*exp(4*x) + 4*x - 15)/16. - _Stefano Spezia_, May 28 2023
%t A353096 LinearRecurrence[{7, -11, 5}, {4, 23, 117}, 23] (* _Amiram Eldar_, Apr 23 2022 *)
%t A353096 nxt[{n_, a_}] := {n + 1, 5 a + 4 - n}; NestList[nxt,{1,4},30][[;;,2]] (* _Harvey P. Dale_, Apr 28 2023 *)
%o A353096 (PARI) my(N=30, x='x+O('x^N)); Vec(x*(4-5*x)/((1-x)^2*(1-5*x)))
%o A353096 (PARI) a(n) = (3*5^(n+1)+4*n-15)/16;
%o A353096 (PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
%o A353096 a(n) = b(n, 5);
%Y A353096 Cf. A064617, A353094, A353095, A353097, A353098, A353099, A353100.
%Y A353096 Cf. A014827.
%K A353096 nonn,easy
%O A353096 1,1
%A A353096 _Seiichi Manyama_, Apr 23 2022