A229685 -id:A229685 - cp's OEIS frontend

A229679 Number of defective 3-colorings of an n X 2 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

0, 2, 36, 360, 2688, 17280, 101376, 559104, 2949120, 15040512, 74711040, 363331584, 1736441856, 8178892800, 38050725888, 175154135040, 798863917056, 3614214979584, 16234976378880, 72464688218112, 321607151124480

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

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

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

Column 2 of A229685.

Cf. A385601.

Empirical: a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3) for n>5.
Empirical g.f.: 2*x^2 - 12*x^3*(3-6*x+8*x^2) / (4*x-1)^3. - R. J. Mathar, Sep 29 2013
Empirical: a(n) = 3*2^(2*n-5)*(3 - 5*n + 2*n^2) for n>2. - Colin Barker, Jun 13 2017
From Enrique Navarrete, Jul 08 2025: (Start)
The above empirical formulas are correct.
a(n) = 3*binomial(2*(n-1),2)*2^(2*n-5) for n >= 3.
a(n) = 3*A385601(2*(n-1)) for n >= 3. (End)

A229680 Number of defective 3-colorings of an n X 3 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 36, 888, 10896, 108000, 959040, 7952256, 62892288, 480370176, 3571983360, 26000455680, 185993957376, 1311390425088, 9133778681856, 62952545157120, 429958047006720, 2913266164432896, 19601071637004288, 131057481860775936

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

Column 3 of A229685.

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

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

Empirical: a(n) = 18*a(n-1) -108*a(n-2) +216*a(n-3) for n>4.
Empirical: G.f. 12*x^2*(10*x+1)*(10*x-3) / (6*x-1)^3. - R. J. Mathar, Sep 29 2013
Empirical a(n) = 2^n*3^(n-3)*(-45 + 4*n + 16*n^2) for n>1. - Colin Barker, Jun 16 2017

A229681 Number of defective 3-colorings of an n X 4 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 360, 10896, 186576, 2700432, 35485776, 437924880, 5169543120, 59031049104, 656886585168, 7159989801744, 76729919248080, 810700618461840, 8463178886657616, 87441690785378832, 895373932606109136

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

Column 4 of A229685.

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

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

Empirical: a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3) for n > 6.
Conjectures from Colin Barker, Jun 16 2017: (Start)
G.f.: 24*x^2*(15 + 49*x - 839*x^2 + 2007*x^3 - 1296*x^4) / (1 - 9*x)^3.
a(n) = 16*3^(2*n-7)*(-449 + 60*n + 256*n^2) for n > 3.
(End)

A229682 Number of defective 3-colorings of an n X 5 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 2688, 108000, 2700432, 58038768, 1138164048, 21063718224, 373936700880, 6435143958672, 108084508966224, 1780281966880656, 28856162624878800, 461471700766361616, 7295948004100520016, 114218818672804436880

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

Column 5 of A229685.

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

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

Empirical: a(n) = 45*a(n-1) - 675*a(n-2) + 2565*a(n-3) + 25272*a(n-4) - 211410*a(n-5) + 3805380*a(n-7) - 8188128*a(n-8) - 14959080*a(n-9) + 70858800*a(n-10) - 85030560*a(n-11) + 34012224*a(n-12) for n > 13.

Empirical g.f.: 48*x^2*(56 - 270*x - 7191*x^2 + 52596*x^3 + 88749*x^4 - 1358532*x^5 + 992412*x^6 + 10940832*x^7 - 16886556*x^8 - 22289904*x^9 + 58471632*x^10 - 30233088*x^11) / ((1 - 15*x + 18*x^2)^3*(1 - 18*x^2)^3). - Colin Barker, Jun 16 2017

A229683 Number of defective 3-colorings of an n X 6 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 17280, 959040, 35485776, 1138164048, 33555543408, 937213830720, 25175909234736, 656711865897408, 16739441074986336, 418806827449472592, 10318185686783039760, 250943458030201728144, 6036117745064710768416

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

Column 6 of A229685.

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

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

Empirical: a(n) = 75*a(n-1) - 2010*a(n-2) + 19486*a(n-3) + 60075*a(n-4) - 2206575*a(n-5) + 7404210*a(n-6) + 70536825*a(n-7) - 435322350*a(n-8) - 601275555*a(n-9) + 8082823950*a(n-10) - 6257717775*a(n-11) - 54520532190*a(n-12) + 92909172825*a(n-13) + 132368667075*a(n-14) - 366499782594*a(n-15) + 470715894135*a(n-17) - 282429536481*a(n-18) for n > 20.

A229684 Number of defective 3-colorings of an n X 7 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 101376, 7952256, 437924880, 21063718224, 937213830720, 39647663129952, 1617006498774912, 64143524707305984, 2489243835565159488, 94897266574306915296, 3564781098414437505600, 132254323024723203273120

Offset: 1

R. H. Hardin, Sep 27 2013

nonn

Column 7 of A229685.

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

R. H. Hardin, Table of n, a(n) for n = 1..172
R. H. Hardin, Empirical recurrence of order 46

Empirical recurrence of order 46 (see link above).

A229678 Number of defective 3-colorings of an n X n 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 2, 888, 186576, 58038768, 33555543408, 39647663129952, 100358466494032656, 559874175467657854896, 7007180500895781669589584

Offset: 1

R. H. Hardin, Sep 27 2013

Diagonal of A229685.

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

cp's OEIS Frontend

A229679 Number of defective 3-colorings of an n X 2 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229680 Number of defective 3-colorings of an n X 3 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229681 Number of defective 3-colorings of an n X 4 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229682 Number of defective 3-colorings of an n X 5 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229683 Number of defective 3-colorings of an n X 6 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229684 Number of defective 3-colorings of an n X 7 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A229678 Number of defective 3-colorings of an n X n 0..2 array connected diagonally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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