A374722 Number of nonisomorphic spanning trees of the nC_5-snake with constant distance between cutpoints.

1, 6, 24, 120, 570, 2850, 14100, 70500, 351750, 1758750, 8790000, 43950000, 219731250, 1098656250, 5493187500, 27465937500, 137329218750, 686646093750, 3433228125000, 17166140625000, 85830691406250, 429153457031250, 2145767226562500, 10728836132812500, 53644180371093750

Offset: 1

Author

Christian Barrientos, Jul 17 2024

Keywords

• nonn
• easy

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.

Examples

			For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake
__ __ __ __ __ __ __ __, __ __ __ __|__ __ __, __ __ __ \/__ __ __,
            __              __           __
__ __ __ __|__ __, __ __ __|__ __, __ __|__ __
                           |            |__
		

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).

Crossrefs

Cf. A374721.

Programs

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 *)

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

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)

Extensions

a(25) corrected by Georg Fischer, Aug 09 2024