A341463 a(n) = (-1)^(n+1) * (3^n+1)/2.

-1, 2, -5, 14, -41, 122, -365, 1094, -3281, 9842, -29525, 88574, -265721, 797162, -2391485, 7174454, -21523361, 64570082, -193710245, 581130734, -1743392201, 5230176602, -15690529805, 47071589414, -141214768241, 423644304722, -1270932914165, 3812798742494, -11438396227481, 34315188682442

Offset

0,2

Comments

Shown by Tutte (he erroneously gave the negative of this sequence) to be the value of the function g(X_n), where X_n is the graph with one vertex and n loops, and g() is the extension to all graphs of the function f(G) defined on trivalent graphs by f(G) =(-1)^n.Q(G), where 2n is the number of vertices of G, and Q(G) is the number of spanning subgraphs of G such that every vertex of G is incident with 2 edges, and obeying the recursions discussed by Tutte in the article.

This sequence is given in balanced ternary representation as (-1), 1(-1), (-1)11, 1(-1)(-1)(-1), (-1)1111, 1(-1)(-1)(-1)(-1)(-1), etc.

References

W. T. Tutte, Some polynomials associated with graphs, Combinatorics, Proceedings of the British Combinatorial Conference. Vol. 13. Cambridge Univ. Press London, 1973.

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (-4,-3).

Formula

a(n) = -4*a(n-1) - 3*a(n-2) for n > 1.

G.f.: -(1 + 2*x)/(1 + 4*x + 3*x^2). - Stefano Spezia, Feb 13 2021

Prog

(Python)
def a(n):
    return (-1)**(n+1) * (3 ** n + 1) // 2

Crossrefs

Absolute values give A007501.

Cf. A076040.

Sequence in context: A123183 A007051 A124302 * A088355 A113485 A360569

Adjacent sequences: A341460 A341461 A341462 * A341464 A341465 A341466

Keyword

sign,easy

Author

Jack W Grahl, Feb 12 2021

Status

approved