A229586 -id:A229586 - cp's OEIS frontend

A229580 Number of defective 3-colorings of an n X 2 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

1, 6, 40, 224, 1152, 5632, 26624, 122880, 557056, 2490368, 11010048, 48234496, 209715200, 905969664, 3892314112, 16642998272, 70866960384, 300647710720, 1271310319616, 5360119185408, 22539988369408, 94557999988736

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 0
  0 0   2 0   0 1   0 2   1 0   2 2   1 2   2 1   0 2   1 2
  1 0   0 2   1 2   1 1   2 1   1 0   0 1   0 0   0 0   0 2

R. H. Hardin, Table of n, a(n) for n = 1..210
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022.

Column 2 of A229586.

Empirical: a(n) = 8*a(n-1) - 16*a(n-2) for n>3.
a(n) = 4*a(n-1) + 4^(n-1) for n > 2. - Gerald Hillier, May 01 2018
a(n) = (2n - 1)*2^(2n - 3) for n > 1 [Gerson W. Barbosa]. - Gerald Hillier, May 02 2018
Empirical g.f.: x*(1 - 2*x + 8*x^2) / (1 - 4*x)^2. - Colin Barker, May 02 2018
a(n) = A002064(2n-2) - A002064(2n-3) for n > 1. - Daniel Forgues, Aug 31 2018
Empirical: a(n) = Integral_{t>0} dt/Beta(n-t,n+t) for n > 1. - Gregory Gerard Wojnar, Feb 10 2024

A229581 Number of defective 3-colorings of an n X 3 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

2, 28, 264, 2160, 16416, 119232, 839808, 5785344, 39191040, 262020096, 1733363712, 11367641088, 74010599424, 478892113920, 3082323787776, 19747769352192, 126009575866368, 801195213717504, 5077997833420800, 32092946307219456

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
..0..1..1....0..1..2....0..0..1....0..1..2....0..1..2....0..1..2....0..1..0
..0..2..1....0..1..2....1..0..2....2..1..0....2..0..1....0..1..0....2..1..0
..0..2..1....0..2..0....2..1..0....2..2..0....2..2..1....0..1..1....2..2..0

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

Column 3 of A229586.

Empirical: a(n) = 12*a(n-1) - 36*a(n-2).
Conjectures from Colin Barker, Sep 19 2018: (Start)
G.f.: 2*(1 + 2*x) / (1 - 6*x)^2.
a(n) = 2^(n + 1) * 3^(n - 1) * (4*n + 3).
(End)

A229582 Number of defective 3-colorings of an n X 4 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

6, 116, 1620, 19764, 224532, 2440692, 25745364, 265720500, 2697594516, 27033340788, 268094978388, 2636009007156, 25732468879380, 249667710249204, 2409688805255892, 23151313964420532, 221538858133842324

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
..0..1..0..1....0..1..1..2....0..1..0..2....0..1..2..1....0..1..2..0
..0..2..1..2....0..2..0..2....0..1..0..2....0..1..0..1....1..0..1..0
..1..0..1..0....0..2..1..0....1..2..1..2....1..2..0..2....1..2..1..0

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

Column 4 of A229586.

Empirical: a(n) = 18*a(n-1) - 81*a(n-2) for n>3.
Conjectures from Colin Barker, Sep 19 2018: (Start)
G.f.: 2*x*(3 + 4*x + 9*x^2) / (1 - 9*x)^2.
a(n) = 4 * 9^(n - 2) * (16*n - 3) for n>1.
(End)

A229583 Number of defective 3-colorings of an n X 5 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

16, 444, 9156, 167364, 2865780, 47091780, 752194836, 11768185764, 181223769204, 2756144066436, 41496027416532, 619575164338788, 9186393069102132, 135397538308042308, 1985403434065601940, 28983255427638399780

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
..0..1..0..2..1....0..1..0..1..0....0..1..0..1..0....0..1..0..1..0
..0..1..1..2..1....0..1..2..1..2....0..1..0..2..1....0..1..2..1..1
..0..2..0..2..1....0..0..2..1..2....2..1..1..2..0....0..1..0..2..0

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

Column 5 of A229586.

Empirical: a(n) = 30*a(n-1) - 261*a(n-2) + 540*a(n-3) - 324*a(n-4).

Empirical g.f.: 4*x*(4 - 9*x + 3*x^2 - 18*x^3) / (1 - 15*x + 18*x^2)^2. - Colin Barker, Sep 19 2018

A229584 Number of defective 3-colorings of an n X 6 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

40, 1620, 49848, 1375152, 35690460, 890824020, 21639043284, 515235810840, 12081465854052, 279877517457936, 6420164883723960, 146080173897129444, 3301145304079911108, 74165789377610689020, 1657889614174445634696

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
..0..1..2..1..0..1....0..1..0..1..0..1....0..1..2..0..1..2....0..1..2..1..0..1
..0..1..2..0..2..1....2..1..2..1..2..0....2..0..2..0..0..2....2..0..2..1..0..1
..0..1..2..1..0..2....2..1..0..2..1..2....2..0..1..2..1..0....2..0..2..2..0..2

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

Column 6 of A229586.

Empirical: a(n) = 50*a(n-1) - 805*a(n-2) + 4662*a(n-3) - 12150*a(n-4) + 14580*a(n-5) - 6561*a(n-6).

Empirical g.f.: 4*x*(10 - 95*x + 262*x^2 + 93*x^3 - 1485*x^4 + 1701*x^5) / (1 - 25*x + 90*x^2 - 81*x^3)^2. - Colin Barker, Sep 19 2018

A229585 Number of defective 3-colorings of an n X 7 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

96, 5724, 264300, 11035044, 435326724, 16551428868, 613195191972, 22285439501940, 798023885879412, 28242628279003332, 990013172442084324, 34429588531063458516, 1189375483329032617620, 40853134231990887545316

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

Column 7 of A229586.

			Some solutions for n=3:
  0 1 2 0 1 0 2      0 1 0 2 1 2 1      0 1 2 0 1 0 1
  0 1 2 0 2 1 0      0 1 0 2 1 2 1      0 1 2 0 1 0 1
  2 1 2 1 2 1 1      0 2 0 2 1 2 2      0 2 1 0 2 0 2

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

Cf. A229586.

Empirical: a(n) = 84*a(n-1) - 2466*a(n-2) + 31428*a(n-3) - 206469*a(n-4) + 750384*a(n-5) - 1513404*a(n-6) + 1574640*a(n-7) - 656100*a(n-8).

A229587 Number of defective 3-colorings of a 2 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 6, 28, 116, 444, 1620, 5724, 19764, 67068, 224532, 743580, 2440692, 7951932, 25745364, 82904796, 265720500, 848179836, 2697594516, 8551948572, 27033340788, 85232507580, 268094978388, 841477302108, 2636009007156, 8242758323964

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

Row 2 of A229586.

			Some solutions for n=3
..0..1..0....0..1..0....0..1..2....0..1..1....0..1..0....0..1..1....0..1..2
..2..1..1....2..2..1....2..1..1....0..2..0....0..1..1....0..2..1....1..0..2

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (6,-9).

LinearRecurrence[{6,-9},{0,6,28,116},30] (* Harvey P. Dale, Mar 06 2023 *)

a(n) = 6*a(n-1) - 9*a(n-2) for n>4.
From Oboifeng Dira, Mar 04 2015: (Start)
a(n) = 4*(3^(n-2) + 4*(2*n-3)*3^(n-4)) for n>2.
G.f.: (x + x^3 + 2*x^4)/(1-3*x)^2 (except for first term).
(End)

A229588 Number of defective 3-colorings of a 3 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 40, 264, 1620, 9156, 49848, 264300, 1374048, 7036116, 35600376, 178380156, 886616784, 4377006372, 21483378600, 104919416268, 510169843584, 2471194910580, 11929478452824, 57413851328028, 275566856985648

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
..0..1..2....0..1..2....0..1..1....0..1..0....0..1..1....0..0..1....0..1..2
..0..2..1....0..1..2....2..0..1....2..0..2....0..2..0....2..0..1....2..1..2
..1..0..1....0..0..2....2..0..1....2..1..0....0..1..2....1..0..2....0..0..2

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

Row 3 of A229586.

Empirical: a(n) = 10*a(n-1) - 29*a(n-2) + 20*a(n-3) - 4*a(n-4) for n>6.

Empirical g.f.: 4*x^2*(10 - 34*x + 35*x^2 - 47*x^3 + 37*x^4) / (1 - 5*x + 2*x^2)^2. - Colin Barker, Sep 19 2018

A229589 Number of defective 3-colorings of a 4 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 224, 2160, 19764, 167364, 1375152, 11035044, 87040260, 677327004, 5213798784, 39777273072, 301217858676, 2266655495148, 16964398022220, 126372108661164, 937513290523896, 6929886363153768, 51058923535878660, 375112065125492964

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

Row 4 of A229586.

			Some solutions for n=3:
  0 1 0   0 0 1   0 1 0   0 1 0   0 1 0   0 1 2   0 0 1
  2 2 1   2 0 1   2 2 0   2 2 1   0 2 2   2 0 2   1 0 2
  1 0 1   1 2 1   0 2 1   0 2 0   0 1 2   1 0 1   2 1 2
  2 0 2   0 2 0   0 2 1   0 2 0   2 0 2   2 2 0   0 1 0

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

Cf. A229586.

Empirical: a(n) = 18*a(n-1) - 111*a(n-2) + 282*a(n-3) - 333*a(n-4) + 180*a(n-5) - 36*a(n-6) for n > 12.

A229590 Number of defective 3-colorings of a 5 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 1152, 16416, 224532, 2865780, 35690460, 435326724, 5230362804, 62078234652, 729497509476, 8501915068020, 98397780931572, 1132075322344044, 12958123181386308, 147663327988334916, 1676107610928542292

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

Row 5 of A229586.

			Some solutions for n=3:
  0 1 0   0 1 0   0 1 0   0 1 2   0 1 2   0 1 0   0 1 2
  0 2 0   2 1 1   1 2 0   0 1 0   0 1 0   0 1 2   2 0 1
  0 2 0   2 0 2   0 2 0   1 2 1   0 1 0   2 1 1   1 2 0
  0 2 1   2 0 1   1 2 0   1 0 2   0 2 0   2 0 1   1 2 0
  2 0 2   1 2 1   1 2 1   2 0 2   1 0 2   1 0 1   2 1 2

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

Cf. A229586.

Empirical: a(n) = 32*a(n-1) - 384*a(n-2) + 2200*a(n-3) - 6494*a(n-4) + 9016*a(n-5) - 816*a(n-6) - 14888*a(n-7) + 18879*a(n-8) - 5464*a(n-9) - 7472*a(n-10) + 8336*a(n-11) - 3648*a(n-12) + 768*a(n-13) - 64*a(n-14) for n > 18.

cp's OEIS Frontend

A229580 Number of defective 3-colorings of an n X 2 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229581 Number of defective 3-colorings of an n X 3 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229582 Number of defective 3-colorings of an n X 4 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229583 Number of defective 3-colorings of an n X 5 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229584 Number of defective 3-colorings of an n X 6 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229585 Number of defective 3-colorings of an n X 7 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229587 Number of defective 3-colorings of a 2 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229588 Number of defective 3-colorings of a 3 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229589 Number of defective 3-colorings of a 4 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A229590 Number of defective 3-colorings of a 5 X n 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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