A207729 -id:A207729 - cp's OEIS frontend

A207724 Number of n X 3 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100, 56595, 122472, 264061, 568695, 1224190, 2633000, 5660226, 12165489, 26141850, 56165805, 120662377, 259206930, 556801480, 1196027864, 2569059663, 5518244160, 11852866905, 25459111647

Offset: 1

R. H. Hardin, Feb 19 2012

Column 3 of A207729.

			Some solutions for n=4:
  1 1 0   1 1 1   1 0 0   1 0 0   0 1 1   0 1 1   0 0 1
  1 0 1   1 1 1   1 0 0   1 1 1   0 1 1   1 1 1   1 1 1
  1 0 0   1 1 0   1 0 0   1 0 0   1 1 1   0 1 1   0 0 1
  1 1 0   1 0 0   1 1 1   1 0 0   0 0 1   0 1 1   0 0 1

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of formula
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-6,0,0,3,1,0,-1).

Cf. A207729.

Configs:= select(A -> A[1..3] <> [0,0,0] and A[4..6] <> [0,0,0] and A[1..3] <> [0,1,0] and A[4..6] <> [0,1,0],
[seq(convert(x,base,2)[1..6],x=2^6..2^7-1)]):
compat:= proc(i,j) local k,col;
   if Configs[i][4..6] <> Configs[j][1..3] then return 0 fi;
   for k from 1 to 3 do
      col:= [Configs[i][k],Configs[i][k+3],Configs[j][k+3]];
      if col = [0,1,1] or col = [1,0,1] then return 0 fi;
   od;
   1
end proc:
T:= Matrix(36,36,compat):
u:= Vector[row](36, 1):
v:= Vector(36,1):
6,seq(u . T^(n-2) . v,n=2..50); # Robert Israel, Mar 05 2018

LinearRecurrence[{3, -2, 3, -6, 0, 0, 3, 1, 0, -1}, {6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100}, 30] (* Jean-François Alcover, May 15 2023, after Robert Israel's confirmed formula *)

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-7) + a(n-8) - a(n-10).
Formula confirmed by Robert Israel, Mar 05 2018 (see link).
G.f.: x*(6 + 18*x - 18*x^2 + 9*x^3 + 7*x^4 + 3*x^5 - 9*x^6 - x^7 - x^8 + x^9) / ((1 - x)*(1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)). - Colin Barker, Mar 05 2018

A207730 Number of 3 X n 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

6, 36, 78, 169, 611, 2209, 6016, 16384, 51840, 164025, 478305, 1394761, 4249238, 12945604, 38516590, 114597025, 345332595, 1040643081, 3114412896, 9320743936, 27998628896, 84105220081, 252138754753, 755885919889, 2268515874246

Offset: 1

R. H. Hardin, Feb 19 2012

nonn

Row 3 of A207729.

			Some solutions for n=4:
..1..0..1..1....0..1..1..1....0..0..1..1....1..1..1..1....0..1..1..1
..0..0..1..1....1..1..1..1....1..1..1..1....0..0..1..1....0..1..1..1
..0..1..1..1....0..0..1..1....0..0..1..1....0..0..1..1....0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A207729.

Empirical: a(n) = 2*a(n-1) + 5*a(n-3) + 22*a(n-4) - 24*a(n-5) - 15*a(n-6) - 36*a(n-8) + 27*a(n-9).

Empirical g.f.: x*(6 + 24*x + 6*x^2 - 17*x^3 - 39*x^4 - 51*x^5 - 9*x^6 - 9*x^7 + 27*x^8) / ((1 - 3*x)*(1 + x - x^2)*(1 + x^2)*(1 + 3*x^2 - 9*x^4)). - Colin Barker, Mar 05 2018

A207723 Number of n X n 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 16, 78, 441, 8575, 277729, 4799749, 93547584, 8978585264, 1701743731081, 92421758243781, 5198831216732241, 3314060538084914409, 4426079717506620317769, 675694617961142291654422

Offset: 1

R. H. Hardin, Feb 19 2012

nonn

Diagonal of A207729.

			Some solutions for n=4
..0..1..1..0....0..0..1..1....1..0..1..1....1..1..1..1....1..1..0..0
..1..1..1..0....0..1..1..1....1..1..0..1....1..1..1..1....1..1..0..1
..0..1..1..1....1..0..1..1....1..0..0..1....1..1..1..1....1..1..0..0
..0..1..1..0....0..0..1..1....0..0..1..1....0..1..1..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..20

Cf. A207729.

A207725 Number of n X 4 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

9, 81, 169, 441, 1225, 2809, 6241, 14161, 31329, 68121, 148225, 321489, 693889, 1495729, 3222025, 6932689, 14907321, 32046921, 68873401, 147987225, 317944561, 683038225, 1467273025, 3151811881, 6770162961, 14542189281, 31235967169

Offset: 1

R. H. Hardin, Feb 19 2012

nonn

Column 4 of A207729.

			Some solutions for n=4:
..1..1..1..0....1..1..0..0....0..0..1..1....1..0..0..1....1..1..0..0
..1..1..0..1....1..1..0..1....1..0..1..1....1..0..0..1....1..1..0..1
..1..1..0..0....1..1..1..0....0..1..1..1....1..1..0..1....1..1..0..0
..0..1..1..0....1..1..0..0....0..0..1..1....1..0..0..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A207729.

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-7) + a(n-8) - a(n-10).

Empirical g.f.: x*(9 + 54*x - 56*x^2 + 69*x^3 + 51*x^4 - 5*x^5 - 45*x^6 - x^8 - x^9) / ((1 - x)*(1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)). - Colin Barker, Jun 25 2018

A207726 Number of nX5 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

15, 225, 611, 2163, 8575, 27931, 88243, 288813, 926949, 2927115, 9278885, 29390445, 92717065, 292337359, 921806095, 2904391209, 9147369231, 28808166663, 90713456051, 285607327515, 899181685831, 2830809676285, 8911643119525

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Column 5 of A207729

			Some solutions for n=4
..1..0..1..1..1....1..0..1..1..1....1..0..1..1..0....0..0..1..1..0
..1..0..1..1..0....0..1..1..1..1....1..0..1..1..1....1..0..1..1..0
..1..0..1..1..0....0..0..1..1..1....0..0..1..1..0....0..1..1..1..1
..0..1..1..0..1....1..0..1..1..1....0..1..1..0..0....0..0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 4*a(n-1) -3*a(n-2) +10*a(n-3) -27*a(n-4) -6*a(n-5) -15*a(n-6) +51*a(n-7) +41*a(n-8) +31*a(n-9) -51*a(n-10) -29*a(n-11) -27*a(n-12) +15*a(n-13) +6*a(n-14) +3*a(n-15) -4*a(n-16) +a(n-17) +a(n-19)

A207727 Number of nX6 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

25, 625, 2209, 10609, 60025, 277729, 1247689, 5890329, 27426169, 125776225, 580858201, 2686867225, 12388803025, 57136775089, 263724358681, 1216770043329, 5612971227241, 25896730187689, 119479087561201, 551206670239681

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Column 6 of A207729

			Some solutions for n=4
..1..1..1..1..0..1....0..0..1..1..0..1....0..1..1..1..1..0....0..0..1..1..0..0
..1..1..1..1..1..1....1..1..1..1..0..0....1..1..1..1..1..1....1..0..1..1..1..1
..1..1..1..0..0..1....0..0..1..1..0..0....0..0..1..1..0..0....0..0..1..1..0..0
..1..1..1..0..0..1....0..0..1..1..0..0....0..0..1..1..0..0....0..1..1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 6*a(n-1) -7*a(n-2) +31*a(n-3) -128*a(n-4) -30*a(n-5) -116*a(n-6) +788*a(n-7) +798*a(n-8) +678*a(n-9) -2778*a(n-10) -1872*a(n-11) -1842*a(n-12) +3336*a(n-13) +1554*a(n-14) +378*a(n-15) -2736*a(n-16) +912*a(n-17) +312*a(n-18) +1590*a(n-19) -540*a(n-20) -228*a(n-21) -222*a(n-22) +210*a(n-23) -86*a(n-24) +24*a(n-25) -38*a(n-26) +8*a(n-27) -a(n-28) -a(n-30) +a(n-31)

A207728 Number of nX7 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

40, 1600, 6016, 33063, 211680, 1029231, 4799749, 23473944, 111527152, 517774120, 2415619129, 11251097760, 52094413065, 240990441237, 1114766044504, 5150494925352, 23781871171576, 109797744696883, 506801727014800

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Column 7 of A207729

			Some solutions for n=4
..1..1..1..1..1..0..1....0..0..1..1..1..1..1....1..0..1..1..1..0..1
..1..0..1..1..0..0..1....1..1..1..1..0..0..1....0..1..1..1..1..0..1
..0..0..1..1..0..0..1....0..0..1..1..0..0..1....0..0..1..1..0..1..1
..0..1..1..0..1..1..0....0..0..1..1..1..0..1....0..0..1..1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 6*a(n-1) -7*a(n-2) +31*a(n-3) -128*a(n-4) -30*a(n-5) -116*a(n-6) +788*a(n-7) +798*a(n-8) +678*a(n-9) -2778*a(n-10) -1872*a(n-11) -1842*a(n-12) +3336*a(n-13) +1554*a(n-14) +378*a(n-15) -2736*a(n-16) +912*a(n-17) +312*a(n-18) +1590*a(n-19) -540*a(n-20) -228*a(n-21) -222*a(n-22) +210*a(n-23) -86*a(n-24) +24*a(n-25) -38*a(n-26) +8*a(n-27) -a(n-28) -a(n-30) +a(n-31)

A207731 Number of 4Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

9, 81, 189, 441, 2163, 10609, 33063, 103041, 418263, 1697809, 5875227, 20331081, 77333859, 294156801, 1057136187, 3799119769, 14123193995, 52502848225, 191262879795, 696752470089, 2569117858827, 9473043664561, 34677617171731

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Row 4 of A207729

			Some solutions for n=4
..1..0..1..1....1..1..1..0....0..0..1..1....1..1..0..1....1..1..0..0
..1..1..1..1....0..1..1..1....0..1..1..1....1..1..0..1....1..1..0..1
..0..0..1..1....0..1..1..0....1..0..1..1....1..0..0..1....1..1..1..0
..0..0..1..1....1..1..0..0....0..0..1..1....1..0..0..1....1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-1) +12*a(n-3) +98*a(n-4) +12*a(n-5) +34*a(n-6) -230*a(n-7) -1980*a(n-8) -508*a(n-9) +340*a(n-10) +840*a(n-11) +9800*a(n-12) +3000*a(n-13) +2000*a(n-15) -10000*a(n-16)

A207732 Number of 5Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

14, 196, 490, 1225, 8575, 60025, 211680, 746496, 4078944, 22287841, 90968949, 371294361, 1833291198, 9052000164, 39384126042, 171355430401, 810254854919, 3831293402161, 17143608557024, 76711252182016, 355989323916384

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Row 5 of A207729

			Some solutions for n=4
..0..1..1..1....1..0..1..1....1..1..1..0....0..0..1..1....0..1..1..1
..1..0..1..1....0..1..1..1....1..1..0..1....1..0..1..1....1..0..1..1
..0..0..1..1....0..0..1..1....1..1..0..0....0..0..1..1....0..0..1..1
..0..0..1..1....0..0..1..1....1..1..1..0....0..0..1..1....0..1..1..1
..1..0..0..1....0..1..1..1....1..0..0..1....0..0..1..1....1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-1) +21*a(n-3) +285*a(n-4) +40*a(n-5) +116*a(n-6) -1248*a(n-7) -17520*a(n-8) -5440*a(n-9) +3712*a(n-10) +13824*a(n-11) +291840*a(n-12) +102400*a(n-13) +131072*a(n-15) -1048576*a(n-16)

A207733 Number of 6Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

21, 441, 1113, 2809, 27931, 277729, 1029231, 3814209, 27996255, 205492225, 919461235, 4114067881, 26135084283, 166026096369, 818513711715, 4035297528025, 23843629466995, 140886430854481, 730455853771371

Offset: 1

R. H. Hardin Feb 19 2012

nonn

Row 6 of A207729

			Some solutions for n=4
..1..0..0..1....1..0..1..1....1..1..0..1....1..0..1..1....0..1..1..0
..1..0..0..1....0..1..1..1....1..0..1..1....0..1..1..1....0..1..1..1
..1..1..0..1....0..0..1..1....1..0..0..1....0..0..1..1....0..1..1..0
..1..0..0..1....1..0..1..1....1..0..0..1....1..0..1..1....1..1..1..0
..1..0..0..1....0..0..1..1....1..1..0..1....0..1..1..1....0..1..1..0
..1..0..0..1....0..0..1..1....1..0..0..1....0..0..1..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-1) +32*a(n-3) +670*a(n-4) +84*a(n-5) +270*a(n-6) -4602*a(n-7) -97188*a(n-8) -30636*a(n-9) +21060*a(n-10) +124488*a(n-11) +4076280*a(n-12) +1423656*a(n-13) +2847312*a(n-15) -37015056*a(n-16)

cp's OEIS Frontend

A207724 Number of n X 3 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207730 Number of 3 X n 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207723 Number of n X n 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207725 Number of n X 4 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207726 Number of nX5 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207727 Number of nX6 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207728 Number of nX7 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207731 Number of 4Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207732 Number of 5Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

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A207733 Number of 6Xn 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.