A384678 Expansion of (1+x) / (1-2*x-4*x^2+2*x^3).

1, 3, 10, 30, 94, 288, 892, 2748, 8488, 26184, 80824, 249408, 769744, 2375472, 7331104, 22624608, 69822688, 215481600, 665004736, 2052290496, 6333636736, 19546425984, 60322817920, 186164066304, 574526552320, 1773063734016, 5471905544704, 16887012920832

Offset: 0

Author

Sean A. Irvine, Jun 05 2025

Keywords

• nonn
• easy
• walk

Comments

Number of walks of length n starting at vertex 0 in the following graph:

2

/|\

0-1-+-3

\|/

4.

Examples

			a(2)=10 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4, 2-4-1, 2-4-2, 2-4-3.
		

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 (2,4,-2).

Crossrefs

Cf. A384677 (vertices 0 and 1), A000244 (missing edge {0,1}), A384633 (missing edge {2,4}), A384640 (missing edge {1,3}).

Programs

Maple

a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);

Mathematica

CoefficientList[Series[(1+x) / (1-2*x-4*x^2+2*x^3), {x, 0, 32}], x]
LinearRecurrence[{2,4,-2},{1,3,10},30] (* Harvey P. Dale, Jul 07 2025 *)