A083217 a(n) = (2*5^n + (-1)^n)/3.

1, 3, 17, 83, 417, 2083, 10417, 52083, 260417, 1302083, 6510417, 32552083, 162760417, 813802083, 4069010417, 20345052083, 101725260417, 508626302083, 2543131510417, 12715657552083, 63578287760417, 317891438802083

Offset: 0

Author

Paul Barry, Apr 23 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.

a(n) is the number of compositions of n when there are 3 types of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,5).

Crossrefs

Cf. A003683, A067411, A082412.

Programs

Magma

[(2*5^n +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Feb 17 2023

Mathematica

LinearRecurrence[{4,5},{1,3},30] (* Harvey P. Dale, Sep 18 2018 *)

Sage

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,3,4,5, lambda n: 0)
[next(it) for i in range(1,24)] # Zerinvary Lajos, Jul 03 2008

Formula

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

G.f.: (1-x)/((1-5*x)*(1+x)).

E.g.f.: (2*exp(5*x) + exp(-x))/3

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*J(n-j+1) where J(n) = A001045(n). - Paul Barry, May 19 2006

a(0)=1, a(n) = 5*a(n-1) - 2 if n is odd, and a(n) = 5*a(n) + 2 if n is even. - Vincenzo Librandi, Nov 18 2010