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A353096
a(1) = 4; for n > 1, a(n) = 5*a(n-1) + 5 - n.
7
4, 23, 117, 586, 2930, 14649, 73243, 366212, 1831056, 9155275, 45776369, 228881838, 1144409182, 5722045901, 28610229495, 143051147464, 715255737308, 3576278686527, 17881393432621, 89406967163090, 447034835815434, 2235174179077153, 11175870895385747
OFFSET
1,1
FORMULA
G.f.: x * (4 - 5*x)/((1 - x)^2 * (1 - 5*x)).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
a(n) = 3*A014827(n) + n.
a(n) = (3*5^(n+1) + 4*n - 15)/16.
a(n) = Sum_{k=0..n-1} (5 - n + k) * 5^k.
E.g.f.: exp(x)*(15*exp(4*x) + 4*x - 15)/16. - Stefano Spezia, May 28 2023
MATHEMATICA
LinearRecurrence[{7, -11, 5}, {4, 23, 117}, 23] (* Amiram Eldar, Apr 23 2022 *)
nxt[{n_, a_}] := {n + 1, 5 a + 4 - n}; NestList[nxt, {1, 4}, 30][[;; , 2]] (* Harvey P. Dale, Apr 28 2023 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(x*(4-5*x)/((1-x)^2*(1-5*x)))
(PARI) a(n) = (3*5^(n+1)+4*n-15)/16;
(PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 5);
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Apr 23 2022
STATUS
approved