A133467 a(n) = a(n-1) + 6*a(n-2) for n >= 2, a(0)=1, a(1)=2.

1, 2, 8, 20, 68, 188, 596, 1724, 5300, 15644, 47444, 141308, 425972, 1273820, 3829652, 11472572, 34450484, 103285916, 309988820, 929704316, 2789637236, 8367863132, 25105686548, 75312865340, 225946984628, 677824176668, 2033506084436, 6100451144444, 18301487651060

Offset: 0

Author

Philippe Deléham, Jan 03 2008

Keywords

• easy
• nonn

Comments

From Sean A. Irvine, Jun 07 2025: (Start)

The number of walks of length n starting at vertex 0 (or, by symmetry, vertex 2 or 3) in the graph K_{1,1,3}:

1---2

/|\ /

0 | X

\|/ \

4---3. (End)

Examples

			a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3. - _Sean A. Irvine_, Jun 07 2025
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (1,6).

Programs

Maple

Digits := 50:
for n from 0 to 40 do round(.8*3^n+.2*(-2)^n) end do;
# Matt C. Anderson, Jul 18 2017

Mathematica

LinearRecurrence[{1,6},{1,2},30] (* Harvey P. Dale, Apr 05 2014 *)

Sage

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,1,6, lambda n: 0); [next(it) for i in range(0,29)] # Zerinvary Lajos, Jul 03 2008

Formula

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

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*5^(n+1-k). - Philippe Deléham, Jan 08 2008

a(n) = (4 * 3^n + (-2)^n) / 5. - Werner Schulte, Dec 18 2024