A100265 -id:A100265 - cp's OEIS frontend

A033507 Number of matchings in graph P_{4} X P_{n}.

1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448

Offset: 0

Per H. Lundow

nonn

			a(1) = 5: the graph is
. o-o-o-o
and the five matchings are
. o o o o
. o-o o o
. o o-o o
. o o o-o
. o-o o-o

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. Phys., 26(1985), 157-167.

Alois P. Heinz, Table of n, a(n) for n = 0..500
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Index entries for linear recurrences with constant coefficients, signature (9,41,-41,-111,91,29,-23,-1,1).

Column 4 of triangle A210662. Row sums of A100265.
For perfect matchings see A005178.
Cf. A033508-A033511.
Bisection (even part) gives A260034.

a:=[1,5,71,823,10012,120465, 1453535,17525619,211351945];; for n in [10..30] do a[n]:=9*a[n-1]+41*a[n-2]-41*a[n-3]-111*a[n-4]+91*a[n-5] +29*a[n-6]-23*a[n-7]-a[n-8]+a[n-9]; od; a; # G. C. Greubel, Oct 26 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) )); // G. C. Greubel, Oct 26 2019

a:=array(0..20,[1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945]):
for j from 9 to 20 do
  a[j]:=9*a[j-1]+41*a[j-2]-41*a[j-3]-111*a[j-4]+91*a[j-5]+
        29*a[j-6]-23*a[j-7]-a[j-8]+a[j-9]
od:
convert(a,list);
# Sergey Perepechko, Apr 24 2013

LinearRecurrence[{9,41,-41,-111,91,29,-23,-1,1},{1,5,71,823,10012,120465, 1453535,17525619,211351945},30] (* Harvey P. Dale, Mar 27 2015 *)

my(x='x+O('x^30)); Vec((1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9)) \\ G. C. Greubel, Oct 26 2019

def A033507_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) ).list()
A033507_list(30) # G. C. Greubel, Oct 26 2019

From Sergey Perepechko, Apr 24 2013: (Start)
a(n) = 9*a(n-1) +41*a(n-2) -41*a(n-3) -111*a(n-4) +91*a(n-5) +29*a(n-6) -23*a(n-7) -a(n-8) +a(n-9).
G.f.: (1-x) * (1 -3*x -18*x^2 +2*x^3 +12*x^4 +x^5 -x^6) / (1 -9*x -41*x^2 +41*x^3 +111*x^4 -91*x^5 -29*x^6 +23*x^7 +x^8 -x^9). (End)

Edited by N. J. A. Sloane, Nov 15 2009

A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

Original entry on oeis.org

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577

Offset: 1

R. J. Mathar, Apr 30 2016

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

			The triangle starts in row n=1 and column m=0 as:
1,3,1;
4,18,7;
4,27,13;
2,32,32;
3,52,64,7;
3,62,133,40;
3,99,269,110,9;
5,152,437,280,48;
5,163,730,669,138,9;
6,258,1243,1318,433,48;
8,343,1823,2670,1239,154,9;
8,408,2949,5240,2849,600,48;
11,632,4577,9011,6655,1927,172,9;
13,746,6287,16184,14697,4930,777,48;
14,971,9928,28135,28805,13089,2669,190,9;
19,1394,14234,44806,58022,32176,7501,954,48;
21,1610,19501,75702,111795,70427,22344,3445,208,9;
25,2224,29785,121302,199354,157078,59859,10576,1131,48;
32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;
35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;
44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;
53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;
60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;
76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016

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A033507 Number of matchings in graph P_{4} X P_{n}.

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