0,2

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.

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

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

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

(Python)

def a(n):

return (-1)**(n+1) * (3 ** n + 1) // 2

Absolute values give A007501.

Cf. A076040.

Sequence in context: A123183 A007051 A124302 * A088355 A113485 A360569

Adjacent sequences: A341460 A341461 A341462 * A341464 A341465 A341466

sign,easy

Jack W Grahl, Feb 12 2021

approved