A181245 -id:A181245 - cp's OEIS frontend

A133129 Number of black/white colorings of a 3 X n rectangle which have no monochromatic 2 by 2 subsquares.

1, 8, 50, 322, 2066, 13262, 85126, 546410, 3507314, 22512862, 144506294, 927561722, 5953863490, 38216853518, 245307588134, 1574588362378, 10107019231634, 64875265300670, 416423472774166, 2672952594083738, 17157235452223586, 110129423550044398

Offset: 0

Victor S. Miller, Sep 19 2007

nonn

			a(2) = 50 because if the middle row is not monochromatic, the top and bottom rows are unconstrained, contributing 2*4*4. if the middle row is monochromatic, the top and bottom rows can each take on only 3 values contributing 2*3*3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, 3, -2).

Cf. A055099, A133357.

Column k=3 of A181245.

G.f.: 1+x*(8+2*x-2*x^2)/(1-6*x-3*x^2+2*x^3). - Colin Barker, Jan 04 2012

More terms from Colin Barker, Jan 03 2012

a(0)=1 prepended and g.f. adapted by Alois P. Heinz, Feb 19 2015

A133130 Number of 0/1 colorings of an n X n square for which no 2 by 2 subsquare is monochromatic.

Original entry on oeis.org

1, 2, 14, 322, 23858, 5735478, 4468252414, 11282914491066, 92343922085798834, 2449629600675855540670, 210618917058297166847778158, 58694743562963266347581955456602, 53015873227026172656988353687982082782, 155209215810704933798454506348361943868443334

Offset: 0

Victor S. Miller, Sep 19 2007

nonn

For each n we define an undirected labeled graph (with self loops), where the vertices are labeled with strings from {0,1}^n and there is an edge between two vertices exactly when we can form a 2 X n rectangle whose rows are the two labels and the 2 X n rectangle has no monochromatic 2 X 2 subsquares. a(n) is the number of walks of length n in this graph. Thus it is the sum of all of the entries of A^n, where A is the adjacency matrix of the graph.

			a(2) = 14 because 2 of the 16 unrestricted colorings are monochromatic.

Alois P. Heinz, Table of n, a(n) for n = 0..15

Cf. A055099.

Main diagonal of A181245.

a(0)-a(1), a(11)-a(13) from Alois P. Heinz, Feb 18 2015

A181239 Number of n X 4 binary matrices with no 2 X 2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

16, 178, 2066, 23858, 275690, 3185462, 36806846, 425288998, 4914052362, 56780001474, 656071271338, 7580653428086, 87591560417790, 1012087088416198, 11694280472399626, 135122952690874754, 1561294205914890890

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Column 4 of A181245.

R. H. Hardin, Table of n, a(n) for n=1..200

Cf. A181245.

Empirical: a(n) = 10*a(n-1) + 20*a(n-2) - 21*a(n-3) - 30*a(n-4) + 8*a(n-5).

Empirical g.f.: 2*x*(8 + 9*x - 17*x^2 - 13*x^3 + 4*x^4) / (1 - 10*x - 20*x^2 + 21*x^3 + 30*x^4 - 8*x^5). - Colin Barker, Mar 26 2018

A181240 Number of nX5 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

32, 634, 13262, 275690, 5735478, 119310334, 2481942354, 51630303190, 1074033301458, 22342450688162, 464776188850718, 9668451712623486, 201126823551119586, 4183916965585685474, 87035437968274038914

Offset: 1

R. H. Hardin Oct 10 2010

nonn

Column 5 of A181245

R. H. Hardin, Table of n, a(n) for n=1..200

Empirical: a(n)=21*a(n-1)+9*a(n-2)-278*a(n-3)+73*a(n-4)+790*a(n-5)-662*a(n-6)+29*a(n-7)+69*a(n-8)-10*a(n-9)

A181241 Number of nX6 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

64, 2258, 85126, 3185462, 119310334, 4468252414, 167341334542, 6267120468434, 234710735573170, 8790181730741270, 329202219837957230, 12328994417789802954, 461734746019197769254, 17292486998904958867138

Offset: 1

R. H. Hardin Oct 10 2010

nonn

Column 6 of A181245

R. H. Hardin, Table of n, a(n) for n=1..200

Empirical: a(n)=36*a(n-1)+120*a(n-2)-2391*a(n-3)-3905*a(n-4)+50702*a(n-5)+27152*a(n-6)-396016*a(n-7)+154999*a(n-8)+751787*a(n-9)-499260*a(n-10)-410368*a(n-11)+355981*a(n-12)+38077*a(n-13)-70276*a(n-14)+6203*a(n-15)+3386*a(n-16)-622*a(n-17)+28*a(n-18)

A181242 Number of nX7 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

128, 8042, 546410, 36806846, 2481942354, 167341334542, 11282914491066, 760744103230314, 51292751478532550, 3458385370907981754, 233179719922207016438, 15722013576638868003178, 1060048065098345660585310

Offset: 1

R. H. Hardin Oct 10 2010

nonn

Column 7 of A181245

R. H. Hardin, Table of n, a(n) for n=1..200

Empirical: a(n)=77*a(n-1)-429*a(n-2)-16791*a(n-3)+132938*a(n-4)+1140609*a(n-5)-11250708*a(n-6)-21101443*a(n-7)+356560316*a(n-8)-276630106*a(n-9)-3595865197*a(n-10)+5253257444*a(n-11)+16399879057*a(n-12)-30419637636*a(n-13)-37486637674*a(n-14)+87632998667*a(n-15)+40083109062*a(n-16)-140235056122*a(n-17)-7589163210*a(n-18)+128111780723*a(n-19)-23221600421*a(n-20)-65939015129*a(n-21)+21868944788*a(n-22)+18307048178*a(n-23)-8259596531*a(n-24)-2431120428*a(n-25)+1497147381*a(n-26)+85285300*a(n-27)-123174410*a(n-28)+8581030*a(n-29)+3300116*a(n-30)-512304*a(n-31)+18304*a(n-32)

A181243 Number of nX8 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

256, 28642, 3507314, 425288998, 51630303190, 6267120468434, 760744103230314, 92343922085798834, 11209291140160525086, 1360654834588676117974, 165164912207734179639646, 20048764401201147437094494

Offset: 1

R. H. Hardin Oct 10 2010

nonn

Column 8 of A181245

R. H. Hardin, Table of n, a(n) for n=1..200

A181244 Number of nX9 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

Original entry on oeis.org

512, 102010, 22512862, 4914052362, 1074033301458, 234710735573170, 51292751478532550, 11209291140160525086, 2449629600675855540670, 535331352762077895206878, 116988976006831708160421286

Offset: 1

R. H. Hardin Oct 10 2010

nonn

Column 9 of A181245

R. H. Hardin, Table of n, a(n) for n=1..200

cp's OEIS Frontend

A133129 Number of black/white colorings of a 3 X n rectangle which have no monochromatic 2 by 2 subsquares.

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A133130 Number of 0/1 colorings of an n X n square for which no 2 by 2 subsquare is monochromatic.

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A181239 Number of n X 4 binary matrices with no 2 X 2 circuit having pattern 0101 in any orientation.

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A181240 Number of nX5 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

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A181241 Number of nX6 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

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A181242 Number of nX7 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

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A181243 Number of nX8 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

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A181244 Number of nX9 binary matrices with no 2X2 circuit having pattern 0101 in any orientation.

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