A353096 - OEIS

%I #30 May 28 2023 22:21:21

%S 4,23,117,586,2930,14649,73243,366212,1831056,9155275,45776369,

%T 228881838,1144409182,5722045901,28610229495,143051147464,

%U 715255737308,3576278686527,17881393432621,89406967163090,447034835815434,2235174179077153,11175870895385747

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11,5).

%F G.f.: x * (4 - 5*x)/((1 - x)^2 * (1 - 5*x)).

%F a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).

%F a(n) = 3*A014827(n) + n.

%F a(n) = (3*5^(n+1) + 4*n - 15)/16.

%F a(n) = Sum_{k=0..n-1} (5 - n + k) * 5^k.

%F E.g.f.: exp(x)*(15*exp(4*x) + 4*x - 15)/16. - _Stefano Spezia_, May 28 2023

%t LinearRecurrence[{7, -11, 5}, {4, 23, 117}, 23] (* _Amiram Eldar_, Apr 23 2022 *)

%t nxt[{n_, a_}] := {n + 1, 5 a + 4 - n}; NestList[nxt,{1,4},30][[;;,2]] (* _Harvey P. Dale_, Apr 28 2023 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(x*(4-5*x)/((1-x)^2*(1-5*x)))

%o (PARI) a(n) = (3*5^(n+1)+4*n-15)/16;

%o (PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);

%o a(n) = b(n, 5);

%Y Cf. A064617, A353094, A353095, A353097, A353098, A353099, A353100.

%Y Cf. A014827.

%K nonn,easy

%O 1,1

%A _Seiichi Manyama_, Apr 23 2022