OFFSET
1,2
COMMENTS
a(n) is the number of spanning trees of the cyclic snake formed with n copies of the cycle on 3 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m.
REFERENCES
Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Snake Graph.
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).
FORMULA
a(n) = 2*3^(n-2) + 3^floor((n-2)/2).
From Stefano Spezia, Jul 20 2024: (Start)
G.f.: x*(1 - 5*x^2)/((1 - 3*x)*(1 - 3*x^2)).
E.g.f.: (2*cosh(3*x) + 3*cosh(sqrt(3)*x) + 2*sinh(3*x) + sqrt(3)*sinh(sqrt(3)*x) - 5)/9. (End)
EXAMPLE
For n=2 the a(2)=3 nonisomorphic spanning trees of 2C_3-snake are:
__ __ __ __, __\__ __, __\/__
MATHEMATICA
LinearRecurrence[{3, 3, -9}, {1, 3, 7}, 30] (* Paolo Xausa, Oct 17 2024 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Christian Barrientos, Jul 17 2024
STATUS
approved