A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.

2, 7, 21, 62, 184, 549, 1643, 4924, 14766, 44291, 132865, 398586, 1195748, 3587233, 10761687, 32285048, 96855130, 290565375, 871696109, 2615088310, 7845264912, 23535794717, 70607384131, 211822152372, 635466457094, 1906399371259, 5719198113753, 17157594341234

Offset

1,1

Links

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Formula

G.f.: x * (2 - 3*x)/((1 - x)^2 * (1 - 3*x)).

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

a(n) = A000340(n-1) + n.

a(n) = (3^(n+1) + 2*n - 3)/4.

a(n) = Sum_{k=0..n-1} (3 - n + k) * 3^k.

E.g.f.: exp(x)*(3*exp(2*x) + 2*x - 3)/4. - Stefano Spezia, May 28 2023

Mathematica

LinearRecurrence[{5, -7, 3}, {2, 7, 21}, 28] (* Amiram Eldar, Apr 23 2022 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(x*(2-3*x)/((1-x)^2*(1-3*x)))
(PARI) a(n) = (3^(n+1)+2*n-3)/4;
(PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 3);

Crossrefs

Cf. A064617, A353095, A353096, A353097, A353098, A353099, A353100.

Cf. A000340.

Sequence in context: A007050 A320811 A005666 * A291411 A159972 A106271

Adjacent sequences: A353091 A353092 A353093 * A353095 A353096 A353097

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 23 2022

Status

approved