A208050 -id:A208050 - cp's OEIS frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4034304, 30847500, 393600, 1050, 8, 0, 0, 6, 739642368, 148039757460, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 02 2012

The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212162 for example. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212195 first at (n,k) = (4,5): A(4,5) = 4034304, A212195(4,5) = 4038432.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4034304, ...
  5,  180,   32940,     30847500,      148039757460, ...
  6,  480,  393600,   3312560640,   286169360240640, ...
  7, 1050, 2735250, 123791435250, 97337270132408250, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068246, 6*A068247.
Rows n=1-15 give: A000007, A000038, A040006, 4*A068271, 5*A068272, 6*A068273, 7*A068274, 8*A068275, 9*A068276, 10*A068277, 11*A068278, 12*A068279, 13*A068280, 14*A068281, 15*A068282.
Cf. A212162, A212195, A208054, A208050.

A208054 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 15, 15, 5, 15, 203, 716, 203, 15, 52, 4140, 83440, 83440, 4140, 52, 203, 115975, 18171918, 112073062, 18171918, 115975, 203, 877, 4213597, 6423127757, 346212384169, 346212384169, 6423127757, 4213597, 877, 4140, 190899322

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1.........1.............2................5................15
...1.........2............15..............203..............4140
...2........15...........716............83440..........18171918
...5.......203.........83440........112073062......346212384169
..15......4140......18171918.....346212384169.18633407199331522
..52....115975....6423127757.2043836452962923
.203...4213597.3376465219485
.877.190899322
...
Some solutions for n=4 k=3
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
..2..3..1....2..3..4....2..3..2....2..3..1....2..3..0....2..3..1....2..3..2
..4..2..4....0..5..0....0..4..0....0..4..5....4..5..3....4..5..3....0..1..4
..0..5..0....1..2..1....1..2..1....5..3..4....0..1..0....0..6..4....2..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)

Columns 1-5 are A000110(n-1), A020557(n-1), A208051, A208052, A208053.

Cf. A207868, A212162, A212163, A208050, A068271.

A208044 Number of n X 3 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

2, 8, 44, 244, 1356, 7540, 41932, 233204, 1296972, 7213172, 40116428, 223109620, 1240835916, 6900974452, 38380133836, 213453141236, 1187130917964, 6602291295860, 36718991727308, 204214611724276, 1135750348251468

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Column 3 of A208050.

			Some solutions for n=4:
  0 1 0   0 1 0   0 1 2   0 1 0   0 1 2   0 1 0   0 1 2
  2 3 2   2 3 1   2 0 3   2 3 1   2 3 0   2 3 1   2 3 0
  1 0 1   1 2 0   3 1 2   0 2 0   0 1 3   0 2 3   0 1 2
  2 3 2   3 1 2   0 3 1   1 3 1   2 0 1   3 1 2   2 3 1

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

Cf. A208050.

Empirical: a(n) = 7*a(n-1) - 8*a(n-2) for n>3.
Conjectures from Colin Barker, Mar 06 2018: (Start)
G.f.: 2*x*(1 - x)*(1 - 2*x) / (1 - 7*x + 8*x^2).
a(n) = (2^(-5-n)*((7-sqrt(17))^n*(-13+5*sqrt(17)) + (7+sqrt(17))^n*(13+5*sqrt(17)))) / sqrt(17) for n>1.
(End)

A208046 Number of n X 4 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

5, 32, 244, 1904, 14976, 118096, 931968, 7356288, 58068800, 458389760, 3618506752, 28564366592, 225486229504, 1779981604864, 14051122574336, 110919150841856, 875592537571328, 6911901929525248, 54562352063938560, 430713614487224320

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Column 4 of A208050.

			Some solutions for n=4:
  0 1 2 1    0 1 2 0    0 1 0 1    0 1 2 3    0 1 2 3
  2 3 0 2    2 0 1 2    2 3 2 3    2 3 0 1    2 3 0 1
  1 2 1 3    1 3 0 1    0 1 0 1    0 1 3 0    1 2 3 2
  3 0 2 0    2 1 2 3    2 3 2 0    2 0 2 1    3 1 0 1

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

Cf. A208050.

Empirical: a(n) = 12*a(n-1) - 40*a(n-2) + 68*a(n-3) - 64*a(n-4) for n>5.

Empirical g.f.: x*(5 - 28*x + 60*x^2 - 84*x^3 + 32*x^4) / (1 - 12*x + 40*x^2 - 68*x^3 + 64*x^4). - Colin Barker, Mar 06 2018

A208047 Number of nX5 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

14, 128, 1356, 14976, 168096, 1897888, 21472544, 243113056, 2753187616, 31181409632, 353155643424, 3999817236576, 45301793546528, 513087070702432, 5811214258319904, 65817707844947552, 745450199930022688

Offset: 1

R. H. Hardin Feb 22 2012

nonn

Column 5 of A208050

			Some solutions for n=4
..0..1..0..1..2....0..1..0..1..0....0..1..0..1..2....0..1..0..1..0
..2..3..2..3..0....2..3..2..3..2....3..2..3..0..3....2..3..2..3..1
..0..1..0..1..2....0..1..0..1..3....1..0..1..2..0....1..0..1..0..2
..3..2..3..0..3....2..3..2..0..1....2..3..0..1..2....2..3..2..1..0

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

Empirical: a(n) = 22*a(n-1) -177*a(n-2) +832*a(n-3) -2664*a(n-4) +5408*a(n-5) -4544*a(n-6) -7936*a(n-7) +24064*a(n-8) -16384*a(n-9) for n>10

A208049 Number of nX7 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

122, 2048, 41932, 931968, 21472544, 502504448, 11838995200, 279733684992, 6617787903744, 156641740131072, 3708483510235904, 87806041737925888, 2079067090247547648, 49228778299543155968, 1165660962446546688768

Offset: 1

R. H. Hardin Feb 22 2012

nonn

Column 7 of A208050

			Some solutions for n=4
..0..1..2..3..1..2..3....0..1..2..3..0..1..2....0..1..0..1..2..3..2
..2..0..1..0..3..1..0....2..0..1..2..3..0..1....2..3..2..0..1..0..3
..3..2..3..2..0..2..1....3..2..3..0..1..2..3....0..1..3..2..3..1..2
..0..1..0..1..3..0..2....0..1..2..3..0..1..0....2..0..1..0..2..3..0

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

Empirical: a(n) = 75*a(n-1) -2554*a(n-2) +55718*a(n-3) -900373*a(n-4) +11540175*a(n-5) -120646224*a(n-6) +1032906704*a(n-7) -7143273200*a(n-8) +38487096736*a(n-9) -148139743168*a(n-10) +301384072128*a(n-11) +434395534080*a(n-12) -4212812674816*a(n-13) -6055937053184*a(n-14) +188862038718976*a(n-15) -1113928611287040*a(n-16) +3311133232840704*a(n-17) -2101437137223680*a(n-18) -32416286559526912*a(n-19) +220131231030444032*a(n-20) -992034773109800960*a(n-21) +3717529494924165120*a(n-22) -11794027245416480768*a(n-23) +31236993687432921088*a(n-24) -70339147326082842624*a(n-25) +154801537571326263296*a(n-26) -448746548849001103360*a(n-27) +1717656393065237053440*a(n-28) -6299441128934805078016*a(n-29) +18664452298317869613056*a(n-30) -42448339422308433133568*a(n-31) +70184786763815892025344*a(n-32) -64759214749331813826560*a(n-33) -65052874836065503936512*a(n-34) +487617470914554383826944*a(n-35) -1485610864045675360813056*a(n-36) +3640029148467676101738496*a(n-37) -8283595658363681883815936*a(n-38) +17664539130299538596691968*a(n-39) -33111040019545470867079168*a(n-40) +49915764398968934926123008*a(n-41) -53360782694992243014500352*a(n-42) +27154724025092859785904128*a(n-43) +27407517485303035101970432*a(n-44) -87711501058877861501337600*a(n-45) +139238385451601130420174848*a(n-46) -222221580176017612232196096*a(n-47) +397056573879679769567559680*a(n-48) -556634782068810820628774912*a(n-49) +453347182355485940514816000*a(n-50) -154742504910672534362390528*a(n-51) for n>52

A208045 Number of n X n 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 2, 44, 1904, 168096, 30818432, 11838995200, 9578237457408, 16372889877477376, 59262411624914853888, 454901302416508366209024, 7413673375795915677473636352

Offset: 1

R. H. Hardin Feb 22 2012

nonn

Diagonal of A208050

			Some solutions for n=4
..0..1..2..1....0..1..2..1....0..1..2..3....0..1..0..1....0..1..2..1
..2..0..3..2....2..3..0..3....3..0..1..0....2..3..2..3....2..3..0..2
..3..2..1..0....0..1..2..0....1..2..3..1....0..1..0..1....0..2..1..3
..0..3..2..1....2..3..1..3....3..1..0..2....2..3..2..0....1..0..2..1

A208050 = Cases[Import["https://oeis.org/A208050/b208050.txt", "Table"], {, }][[All, 2]];
a[n_] := A208050[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Sep 23 2019 *)

A208048 Number of nX6 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

41, 512, 7540, 118096, 1897888, 30818432, 502504448, 8206614784, 134107752704, 2192020679040, 35832132655104, 585752326535168, 9575474013944320, 156533905282970624, 2558923378953086464, 41831786939400151040

Offset: 1

R. H. Hardin Feb 22 2012

nonn

Column 6 of A208050

			Some solutions for n=4
..0..1..2..0..2..3....0..1..2..0..1..3....0..1..2..1..3..0....0..1..2..3..0..1
..3..0..1..3..0..2....2..0..1..2..0..1....3..0..3..2..1..3....3..0..1..2..3..2
..1..2..0..2..3..0....3..2..3..1..2..3....1..2..1..3..0..2....1..2..0..1..0..3
..3..1..3..0..1..2....1..0..2..3..0..1....3..0..2..1..3..1....3..1..2..3..1..0

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

Empirical: a(n) = 40*a(n-1) -672*a(n-2) +7048*a(n-3) -53920*a(n-4) +315080*a(n-5) -1388592*a(n-6) +4290880*a(n-7) -7204704*a(n-8) -5415680*a(n-9) +63216624*a(n-10) -131003904*a(n-11) -9639424*a(n-12) +571912192*a(n-13) -1570279424*a(n-14) +4233822208*a(n-15) -11382816768*a(n-16) +22231908352*a(n-17) -31692161024*a(n-18) +37178310656*a(n-19) -34628173824*a(n-20) +17179869184*a(n-21) for n>22

cp's OEIS Frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

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A208054 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208044 Number of n X 3 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208046 Number of n X 4 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208047 Number of nX5 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208049 Number of nX7 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208045 Number of n X n 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208048 Number of nX6 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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