A374722 - OEIS

%I #32 Sep 03 2024 00:37:12

%S 1,6,24,120,570,2850,14100,70500,351750,1758750,8790000,43950000,

%T 219731250,1098656250,5493187500,27465937500,137329218750,

%U 686646093750,3433228125000,17166140625000,85830691406250,429153457031250,2145767226562500,10728836132812500,53644180371093750

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

%C 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.

%D Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-25).

%F a(n) = (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2)) for n > 1.

%F From _Stefano Spezia_, Jul 23 2024: (Start)

%F G.f.: x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)).

%F 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)

%e For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake

%e __ __ __ __ __ __ __ __, __ __ __ __|__ __ __, __ __ __ \/__ __ __,

%e             __              __           __

%e __ __ __ __|__ __, __ __ __|__ __, __ __|__ __

%e                            |            |__

%t 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 *)

%o (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

%Y Cf. A374721.

%K nonn,easy

%O 1,2

%A _Christian Barrientos_, Jul 17 2024

%E a(25) corrected by _Georg Fischer_, Aug 09 2024