OFFSET
1,2
COMMENTS
a(n) is the number of spanning trees of the cyclic snake formed with n copies of the cycle on 5 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
Index entries for linear recurrences with constant coefficients, signature (5,5,-25).
FORMULA
a(n) = (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2)) for n > 1.
From Stefano Spezia, Jul 23 2024: (Start)
G.f.: x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)).
E.g.f.: (24 + 10*x - 9*cosh(5*x) - 15*cosh(sqrt(5)*x) - 9*sinh(5*x) - 3*sqrt(5)*sinh(sqrt(5)*x))/50. (End)
EXAMPLE
For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake
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MATHEMATICA
Drop[CoefficientList[Series[x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)), {x, 0, 30}], x], 1] (* Georg Fischer, Aug 09 2024 *)
PROG
(PARI) for(n=1, 30, print1(if(n==1, 1, (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2))), ", ")) \\ Georg Fischer, Aug 09 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Christian Barrientos, Jul 17 2024
EXTENSIONS
a(25) corrected by Georg Fischer, Aug 09 2024
STATUS
approved