A102089 -id:A102089 - cp's OEIS frontend

A102091 Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices).

12, 32, 108, 392, 1452, 5408, 20172, 75272, 280908, 1048352, 3912492, 14601608, 54493932, 203374112, 759002508, 2832635912, 10571541132, 39453528608, 147242573292, 549516764552, 2050824484908, 7653781175072, 28564300215372, 106603419686408, 397849378530252

Offset: 1

Emeric Deutsch, Dec 29 2004

Conjecture: For an integer n > 6, primes of the form either 12k+5 or 12k+7 will never appear in factors of any centered n-gonal number, if and only if n is in A102091. - Steven Lu, Nov 15 2022

			a(1)=12 because in the graph C_2 X P_3 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,A'B',B'C',a,a',b,b',c,c'}, where a and a' are two edges between A and A', b and b' are two edges between B and B' and c and c' are two edges between C and C', we have the following twelve perfect matchings: eight matchings by taking one edge from each of the pairs {a,a'},{b,b'} and {c,c'}; two matchings by taking AB, A'B' and either edge from the pair {c,c'}; two matchings by taking BC, B'C' and either edge from the pair {a,a'}.

Colin Barker, Table of n, a(n) for n = 1..1000
H. Hosoya and A. Motoyama, 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, J. Math. Physics 26 (1985) 157-167 (eq. (53) and Table VII).
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A102089.

a:=[12,32,108];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2] + a[n-3]; od; a; # G. C. Greubel, Oct 27 2019

[Floor(2*(2+(2-Sqrt(3))^n+(2+Sqrt(3))^n)): n in [1..25]]; // Vincenzo Librandi, Nov 04 2016

a[1]:=12: a[2]:=32: a[3]:=108: for n from 4 to 31 do a[n]:=5*a[n-1] -5*a[n-2]+a[n-3] od:seq(a[n],n=1..25);

CoefficientList[Series[4(3-x)(1-2x)/((1-x)(1-4x+x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 04 2016 *)
LinearRecurrence[{5,-5,1}, {12,32,108}, 31] (* G. C. Greubel, Oct 27 2019 *)

Vec(4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 03 2016

def A102091_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)) ).list()
a=A102091_list(30); a[1:] # G. C. Greubel, Oct 27 2019

a(n) = A102089(2*n, 3*n).
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1)=12, a(2)=32 and a(3)=108.
G.f.: 4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)).
a(n) = A001353(n+1) - 7*A001353(n-1) + 4. - Ralf Stephan, May 17 2007
a(n) = 4*A102206(n-1). - R. J. Mathar, Mar 06 2016
a(n) = 2*(2 + (2-sqrt(3))^n + (2+sqrt(3))^n). - Colin Barker, Nov 03 2016
E.g.f.: -8 + 4*exp(x)*(1 + exp(x)*cosh(sqrt(3)*x)). - Stefano Spezia, Oct 27 2019

A102090 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).

Original entry on oeis.org

47, 228, 1511, 9213, 57536, 356863, 2217871, 13775700, 85579087, 531616825, 3302453192, 20515048427, 127440964999, 791672146068, 4917923140383, 30550483740725, 189781751728736, 1178937572877255, 7323643025265351

Offset: 2

Emeric Deutsch, Dec 29 2004

nonn

Row sums of A102089.

G. C. Greubel, Table of n, a(n) for n = 2..1000
H. Hosoya and A. Motoyama, 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, J. Math. Physics 26 (1985) 157-167 (eq. (52) and Table VII).
Index entries for linear recurrences with constant coefficients, signature (3,19,10,-24,-10,11,1,-1).

Column 3 of A287428.

Cf. A102089.

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

R:=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

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);

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

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

def A102090_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    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()
A102090_list(30) # G. C. Greubel, Oct 27 2019

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.

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)).

cp's OEIS Frontend

A102091 Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices).

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