A100245 -id:A100245 - cp's OEIS frontend

A033506 Number of matchings in graph P_{3} X P_{n}.

1, 3, 22, 131, 823, 5096, 31687, 196785, 1222550, 7594361, 47177097, 293066688, 1820552297, 11309395995, 70254767718, 436427542283, 2711118571311, 16841658983944, 104621568809247, 649916534985369, 4037327172325542

Offset: 0

Per H. Lundow

nonn

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 50, 999.
Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research reports, No 12, 1996, Department of Mathematics, Umea University.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Svenja Huntemann, Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. C. Read, The dimer problem for narrow rectangular arrays: A unified method of solution, and some extensions, Aequationes Mathematicae, 24 (1982), 47-65.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Index entries for linear recurrences with constant coefficients, signature (4,14,0,-10,0,1).

Column 3 of triangle A210662. Row sums of A100245.

Cf. A030186, A033507, A033508, A033509, A033510, A033511.

a:=[1,3,22,131,823,5096];; for n in [7..30] do a[n]:=4*a[n-1] +14*a[n-2]-10*a[n-4]+a[n-6]; od; a; # G. C. Greubel, Oct 26 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)) )); // G. C. Greubel, Oct 26 2019

seq(coeff(series((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5 )), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 26 2019

CoefficientList[Series[(1-2x-x^2)(1+x-x^2)/((1+x)(1-5x-9x^2+9x^3+x^4-x^5) ), {x, 0, 30}], x] (* Harvey P. Dale, Dec 05 2014 *)
LinearRecurrence[{4, 14, 0, -10, 0, 1}, {1, 3, 22, 131, 823, 5096}, 30] (* Harvey P. Dale, Dec 05 2014 *)
Table[RootSum[-1 +# +9#^2 -9#^3 -5#^4 +#^5 &, 1436541#^n + 3941068#^(n+1) -6086452#^(n+2) -2800519#^(n+3) +591744#^(n+4) &]/10204570 -(-1)^n/5, {n, 20}] (* Eric W. Weisstein, Oct 02 2017 *)

my(x='x+O('x^30)); Vec((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2 +9*x^3+x^4-x^5))) \\ G. C. Greubel, Oct 26 2019

def A033506_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)) ).list()
A033506_list(30) # G. C. Greubel, Oct 26 2019

G.f.: (1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)). - Sergey Perepechko, Apr 19 2013

A272472 Triangle T(n,m) by rows: The number of tatami tilings of a 3 by n grid with dimers and m monomers.

Original entry on oeis.org

0, 2, 0, 1, 3, 0, 9, 0, 1, 0, 10, 0, 12, 4, 0, 27, 0, 13, 0, 18, 0, 56, 0, 16, 6, 0, 75, 0, 97, 0, 18, 0, 38, 0, 198, 0, 152, 0, 18, 10, 0, 177, 0, 433, 0, 214, 0, 18, 0, 72, 0, 570, 0, 836, 0, 282, 0, 18, 16, 0, 393, 0, 1517, 0, 1442, 0, 354, 0, 18, 0, 136

Offset: 1

R. J. Mathar, Apr 30 2016

			The triangle starts in row n=1 and column m=0 as:
0,2,0,1;
3,0,9,0,1;
0,10,0,12;
4,0,27,0,13;
0,18,0,56,0,16;
6,0,75,0,97,0,18;
0,38,0,198,0,152,0,18;
10,0,177,0,433,0,214,0,18;
0,72,0,570,0,836,0,282,0,18;
16,0,393,0,1517,0,1442,0,354,0,18;
0,136,0,1478,0,3472,0,2292,0,426,0,18;
26,0,829,0,4571,0,7052,0,3410,0,498,0,18;
0,250,0,3554,0,12070,0,13076,0,4808,0,570,0,18;
42,0,1691,0,12479,0,28158,0,22480,0,6494,0,642,0,18;
0,454,0,8108,0,37222,0,59530,0,36308,0,8468,0,714,0,18;
68,0,3359,0,31729,0,97766,0,115948,0,55672,0,10730,0,786,0,18;
0,814,0,17768,0,105238,0,231622,0,210880,0,81708,0,13280,0,858,0,18;
110,0,6537,0,76483,0,306606,0,503348,0,361878,0,115568,0,16118,0,930,0,18;
0,1446,0,37736,0,278626,0,803060,0,1016880,0,590846,0,158404,0,19244,0,1002,0,18;
178,0,12511,0,176833,0,889916,0,1923278,0,1929730,0,924216,0,211368,0,22658,0,1074,0,18;
0,2548,0,78144,0,700670,0,2549216,0,4268026,0,3469042,0,1392996,0,275612,0,26360,0,1146,0,18;
288,0,23617,0,395387,0,2430464,0,6661414,0,8867630,0,5948792,0,2032802,0,352288,0,30350,0,1218,0,18;
0,4460,0,158492,0,1690478,0,7547920,0,16089358,0,17395888,0,9787628,0,2883858,0,442548,0,34628,0,1290,0,18;
466,0,44067,0,860069,0,6319840,0,21344172,0,36292416,0,32446518,0,15527142,0,3990996,0,547544,0,39194,0,1362,0,18;

A. Erickson, F. Ruskey, J. Woodcock, M. Schurch, Auspicious tatami mat arrangements, arXiv:1103.3309 (2011).

Cf. A180970 (row sums), A068922 (column m=0), A271786 (column m=1), A272471 (2 by n grid), A100245 (row reversed without tatami condition).

G.f. x *(x^4*y^3 +7*x*y^2 +3*x +2*y +y^3 +x^6*y +3*x^2*y -x^3*y^2 -6*x^4*y -x^2*y^5 +x^2*y^3 +y^3*x^6 -2*y^4*x^5 -2*x^3 -2*x^5 +y^5*x^4 -x^3*y^4 -x^5*y^2 +x^7) / (x^6 +x^5*y -2*x^4*y^2 -2*x^2 -x*y +1). - R. J. Mathar, May 01 2016

cp's OEIS Frontend

A033506 Number of matchings in graph P_{3} X P_{n}.

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