A102090 - OEIS

%I #17 Sep 08 2022 08:45:16

%S 47,228,1511,9213,57536,356863,2217871,13775700,85579087,531616825,

%T 3302453192,20515048427,127440964999,791672146068,4917923140383,

%U 30550483740725,189781751728736,1178937572877255,7323643025265351

%N Number of matchings in the C_n X P_3 graph (C_n is the cycle graph on n vertices and P_3 is the path graph on 3 vertices).

%C Row sums of A102089.

%H G. C. Greubel, <a href="/A102090/b102090.txt">Table of n, a(n) for n = 2..1000</a>

%H H. Hosoya and A. Motoyama, <a href="http://dx.doi.org/10.1063/1.526778">An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices</a>, J. Math. Physics 26 (1985) 157-167 (eq. (52) and Table VII).

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,19,10,-24,-10,11,1,-1).

%F a(n) = 3*a(n-1) + 19*a(n-2) + 10*a(n-3) - 24*a(n-4) - 10*a(n-5) + 11*a(n-6) + a(n-7) - a(n-8) with a(2)=47, a(3)=228, a(4)=1511, a(5)=9213, a(6)=57536, a(7)=356863, a(8)=2217871 and a(9)=13775700.

%F G.f.: x^2*(47+87*x-66*x^2-122*x^3+36*x^4+40*x^5-5*x^6-3*x^7)/((1+x)*(1+x-x^2)*(1-5*x-9*x^2+9*x^3+x^4-x^5)).

%p a[2]:=47: a[3]:=228: a[4]:=1511: a[5]:=9213: a[6]:=57536: a[7]:=356863: a[8]:=2217871: a[9]:=13775700: for n from 10 to 23 do a[n]:=3*a[n-1] +19*a[n-2]+10*a[n-3]-24*a[n-4]-10*a[n-5]+11*a[n-6]+a[n-7]-a[n-8] od: seq(a[n],n=2..23);

%t LinearRecurrence[{3,19,10,-24,-10,11,1,-1}, {47,228,1511,9213,57536, 356863,2217871,13775700}, 30] (* _Harvey P. Dale_, Oct 24 2011 *)

%o (PARI) my(x='x+O('x^30)); Vec(x^2*(47+87*x-66*x^2-122*x^3+36*x^4+40*x^5 -5*x^6-3*x^7)/((1+x)*(1+x-x^2)*(1-5*x-9*x^2+9*x^3+x^4-x^5))) \\ _G. C. Greubel_, Oct 27 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x^2*(47+87*x-66*x^2-122*x^3+36*x^4+40*x^5-5*x^6-3*x^7)/((1+x)*(1+x-x^2)*(1-5*x-9*x^2+9*x^3+x^4-x^5)) )); // _G. C. Greubel_, Oct 27 2019

%o (Sage)

%o def A102090_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P(x^2*(47+87*x-66*x^2-122*x^3+36*x^4+40*x^5-5*x^6-3*x^7)/( (1+x)*(1+x-x^2)*(1-5*x-9*x^2+9*x^3+x^4-x^5))).list()

%o A102090_list(30) # _G. C. Greubel_, Oct 27 2019

%o (GAP) a:=[47,228,1511,9213,57536, 356863,2217871,13775700];; for n in [9..30] do a[n]:=3*a[n-1]+19*a[n-2]+10*a[n-3]-24*a[n-4]-10*a[n-5] + 11*a[n-6]+a[n-7]-a[n-8]; od; a; # _G. C. Greubel_, Oct 27 2019

%Y Column 3 of A287428.

%Y Cf. A102089.

%K nonn

%O 2,1

%A _Emeric Deutsch_, Dec 29 2004