A080961 Fourth binomial transform of A010686 (period 2: repeat 1,5).

1, 9, 57, 321, 1713, 8889, 45417, 230001, 1158753, 5820009, 29178777, 146130081, 731358993, 3658920729, 18300980937, 91524036561, 457677578433, 2288560079049, 11443316955897, 57218134461441, 286095321353073, 1430490553902969, 7152494610927657, 35762598578876721

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-15).

Formula

a(n) = 5*a(n-1) + 4*3^(n-1).

a(n) = 3*5^n - 2*3^n.

G.f.: (1+x)/((1-3*x)*(1-5*x)). - Klaus Brockhaus, Nov 26 2009

From Mario C. Enriquez, Dec 08 2016: (Start)

a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.

a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)

a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - Wesley Ivan Hurt, Dec 08 2016

E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 2024

Example

G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...

Maple

A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2016

Mathematica

CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 07 2012 *)

Prog

(Magma) binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;
binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // Klaus Brockhaus, Nov 26 2009
(Magma) [3*5^n - 2*3^n: n in [0..30]]; // Vincenzo Librandi, Dec 07 2012

Crossrefs

Cf. A003948, A005059, A010686, A080960, A080962.

Sequence in context: A014916 A045635 A026896 * A163919 A262490 A180028

Adjacent sequences: A080958 A080959 A080960 * A080962 A080963 A080964

Keyword

easy,nonn

Author

Paul Barry, Mar 03 2003

Extensions

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009

Status

approved