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

%I A374722 #32 Sep 03 2024 00:37:12
%S A374722 1,6,24,120,570,2850,14100,70500,351750,1758750,8790000,43950000,
%T A374722 219731250,1098656250,5493187500,27465937500,137329218750,
%U A374722 686646093750,3433228125000,17166140625000,85830691406250,429153457031250,2145767226562500,10728836132812500,53644180371093750
%N A374722 Number of nonisomorphic spanning trees of the nC_5-snake with constant distance between cutpoints.
%C A374722 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 A374722 Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.
%H A374722 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-25).
%F A374722 a(n) = (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2)) for n > 1.
%F A374722 From _Stefano Spezia_, Jul 23 2024: (Start)
%F A374722 G.f.: x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)).
%F A374722 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 A374722 For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake
%e A374722 __ __ __ __ __ __ __ __, __ __ __ __|__ __ __, __ __ __ \/__ __ __,
%e A374722             __              __           __
%e A374722 __ __ __ __|__ __, __ __ __|__ __, __ __|__ __
%e A374722                            |            |__
%t A374722 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 A374722 (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 A374722 Cf. A374721.
%K A374722 nonn,easy
%O A374722 1,2
%A A374722 _Christian Barrientos_, Jul 17 2024
%E A374722 a(25) corrected by _Georg Fischer_, Aug 09 2024