A068305 -id:A068305 - cp's OEIS frontend

A068239 1/2 the number of colorings of a 3 X 3 square array with n colors.

1, 123, 4806, 71410, 583455, 3232341, 13675228, 47502036, 141991245, 377162335, 910842306, 2033854758, 4253012491, 8411348505, 15856955640, 28673921896, 49991146713, 84387303171, 138412872190, 221253017370, 345558093111, 528471784093, 792890261076

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> (79+(-323+(594+(-648+(459+(-216+(66+(-12+n)*n)*n) *n)*n)*n)*n)*n) *n/2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012: (Start)
G.f.: x^2*(1199*x^7 +16567*x^6 +60099*x^5 +71075*x^4 +28765*x^3 +3621*x^2 +113*x+1) / (x-1)^10.
a(n) = (79*n -323*n^2 +594*n^3 -648*n^4 +459*n^5 -216*n^6 +66*n^7 -12*n^8 +n^9) / 2.
(End)

A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 24, 5, 0, 0, 0, 72, 120, 6, 0, 0, 0, 168, 6720, 360, 7, 0, 0, 0, 360, 935040, 126360, 840, 8, 0, 0, 0, 744, 325061760, 265035240, 1128960, 1680, 9, 0, 0, 0, 1512, 283192323840, 3322711053720, 17160407040, 6510000, 3024, 10

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges; see A212208 for example. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

This graph is also called the king graph. - Andrew Howroyd, Jun 25 2017

			Square array A(n,k) begins:
  1,   0,       0,           0,                0, ...
  2,   0,       0,           0,                0, ...
  3,   0,       0,           0,                0, ...
  4,  24,      72,         168,              360, ...
  5, 120,    6720,      935040,        325061760, ...
  6, 360,  126360,   265035240,    3322711053720, ...
  7, 840, 1128960, 17160407040, 2949948395735040, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, King Graph
Wikipedia, Chromatic Polynomial

Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.
Rows n=1-16 give: A000007, A000038, 3*A000007, 4*A068293, 5*A068294, 6*A068295, 7*A068296, 8*A068297, 9*A068298, 10*A068299, 11*A068300, 12*A068301, 13*A068302, 14*A068303, 15*A068304, 16*A068305.
Cf. A000290, A002943, A212208, A208021.

A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

Original entry on oeis.org

1, 673072, 24674450670, 47695073906240, 16222886703881375, 1842996310592836896, 98798502888215704812, 3068393794369671728640, 62960689505171989129005, 933100312771109288146000, 10639781342848431789710266, 97779035987698387480546752, 750090455960001686602653035

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Alois P. Heinz, Formula for A068248

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a:= n-> (3006792824+ (-26845691044+ (115537440058+ (-319333174471+ (636781496832+ (-975359012827+ (1192518013138+ (-1193724499144+ (995462037353+ (-699932345254+ (418375639535+ (-213720396671+ (93568819963+ (-35133625647+ (11298632584+
(-3101089710+ (722137763+ (-141421592+ (23000726+ (-3051871+ (321994+ (-25992+ (1508+(-56+n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n/6:
seq (a(n), n=3..40);  # Alois P. Heinz, May 03 2012

Extended beyond a(10) by Alois P. Heinz, May 03 2012

A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

Original entry on oeis.org

1, 6, 18, 42, 90, 186, 378, 762, 1530, 3066, 6138, 12282, 24570, 49146, 98298, 196602, 393210, 786426, 1572858, 3145722, 6291450, 12582906, 25165818, 50331642, 100663290, 201326586, 402653178, 805306362, 1610612730, 3221225466, 6442450938, 12884901882

Offset: 1

R. H. Hardin, Feb 24 2002

1/4 the number of colorings of an n X n octagonal array with 4 colors.
Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - N. J. A. Sloane, Feb 20 2016
From Gary W. Adamson, Jan 13 2009: (Start)
Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).
Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)
For n > 1, number of ternary strings of length n with exactly 2 different digits. - Enrique Navarrete, Nov 20 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ana Rechtman, Février 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A068239-A068305, A000332, A002417, A027441.

Cf. A091344. - Gary W. Adamson, Jan 13 2009

[1] cat [6*(2^(n-1)-1): n in [2..40]]; // Vincenzo Librandi, Feb 20 2016

a=0; lst={1}; k=6; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
Transpose[NestList[{First[#]+1,6(2^First[#]-1)}&,{1,1},30]][[2]] (* or *) Join[{1},LinearRecurrence[{3,-2},{6,18},30]] (* Harvey P. Dale, Nov 27 2011 *)

a(n)=polcoeff(prod(i=1,2,(1+i*x))/(prod(i=1,2,(1-i*x))+x*O(x^n)),n)
for(n=0,50,print1(a(n),","))

G.f.: (1+x)*(1+2*x)/((1-x)*(1-2*x)). - Benoit Cloitre, Apr 13 2002
a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Nov 27 2011
E.g.f.: 1 - 6*exp(x)*(exp(x) - 1). - Stefano Spezia, May 18 2024

More terms from Benoit Cloitre, Apr 13 2002

Old definition (which is now a comment) replaced with explicit formula by N. J. A. Sloane, May 12 2010

A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

Original entry on oeis.org

1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396, 31191658416342674, 169426507164530254380, 2176592549084872196370724, 66158464020552857153017287240, 4759146677426447759184119036493676, 810410082813497381147177065840601910384

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..24

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.
See A047938 for number of improper colorings.
Main diagonal of A078099.
Twice A207993 for n>1.

M[1] = {{1}}; M[m_] := M[m] = {{M[m - 1], Transpose[M[m - 1]]}, {Array[0 &, {2^(m - 2), 2^(m - 2)}], M[m - 1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]]; T[m_, 1] := 2^(m - 1); T[1, n_] := 2^(n - 1); T[m_, n_] := MatrixPower[W[m], n - 1] // Flatten // Total; a[n_] := T[n, n]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 01 2017, after code from A078099 *)

For formula see A078099.

More terms from Vladeta Jovovic, Jul 22 2004
a(11)-a(12) from Alois P. Heinz, Mar 25 2009
a(13)-a(14) from Andrew Howroyd, Jun 26 2017

A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

Original entry on oeis.org

1, 12, 264, 11424, 1008576, 184910592, 71033971200, 57469424744448, 98237339264864256, 355574469749489123328, 2729407814499050197254144, 44482040254775494064841818112, 1540473331004371306422199656382464, 113440401780206156918876627438624833536

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Andrew Howroyd, Table of n, a(n) for n = 1..16

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a(9) from Alois P. Heinz, May 02 2012

a(10)-a(14) from Andrew Howroyd, Jun 25 2017

A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 176, 5490, 65600, 455875, 2239776, 8647716, 27962880, 78920325, 200002000, 464447126, 1003294656, 2039332295, 3935444800, 7261533000, 12884914176, 22089914121, 36733221360, 59442494650, 93866696000, 144987663051, 219503536736, 326295822700, 476993088000

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a:= n-> (248 +(-1012 +(1786 +(-1791 +(1120 +(-448 +(112 +(-16+n)*n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40);  #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: (1089*x^6+10934*x^5+26015*x^4+18500*x^3+3775*x^2+166*x+1) / (x-1)^10*x^3.
a(n) = (n^9 -16*n^8 +112*n^7 -448*n^6 +1120*n^5 -1791*n^4 +1786*n^3 -1012*n^2 +248*n)/6.  (End)

A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 7616, 5141250, 552093440, 20631905875, 395001645696, 4771909547076, 41190314035200, 275192443300005, 1502690499112000, 6971521964029766, 28275884687022336, 102456840191225975, 337289521082456320, 1022310183284613000, 2883605488481550336, 7636012822945480521

Offset: 3

R. H. Hardin, Feb 24 2002

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

[(n^11 -26*n^10 +310*n^9 -2240*n^8 +10915*n^7 -37726*n^6 +94576*n^5 -172395*n^4 +224588*n^3 -199854*n^2 +109788*n -28340)*n *(n-1)*(n-2)^3/6: n in [3..19]]; // Bruno Berselli, May 03 2012

a:= n-> (-226720+ (1445104+ (-4304712+ (7968348+ (-10265148+ (9755858+ (-7068408+ (3975561+ (-1749715+ (602408+ (-160859+ (32703+ (-4898+ (510+ (-33+n)*n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40); #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: -(7926831*x^13 +710120929*x^12 +16477733814*x^11 +144915014346*x^10 +569769493505*x^9 +1086745824783*x^8 +1040642122932*x^7 +499586289612*x^6 +115866023553*x^5 +11940350895*x^4 +465727286*x^3 +5011914*x^2 +7599*x+1) *x^3 / (x-1)^17.
a(n) = (n^16 -33*n^15 +510*n^14 -4898*n^13 +32703*n^12 -160859*n^11 +602408*n^10 -1749715*n^9  +3975561*n^8 -7068408*n^7 +9755858*n^6 -10265148*n^5 +7968348*n^4 -4304712*n^3 +1445104*n^2 -226720*n)/6. (End)

Extended beyond a(15) by Alois P. Heinz, May 02 2012

A068254 1/4 the number of colorings of an n X n square array with 4 colors.

Original entry on oeis.org

1, 21, 2403, 1500183, 5110723191, 95013316876491, 9639473169171326643, 5336900216006709884938623, 16124704040675904181778734982451, 265865038636937159336134567410478299051

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..16

Main diagonal of A222444.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222444 = Cases[Import["https://oeis.org/A222444/b222444.txt", "Table"], {, }][[All, 2]];
a[n_] := A222444[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Sep 23 2019 *)

a(9)-a(10) from Alois P. Heinz, Apr 27 2012

A068255 1/5 the number of colorings of an n X n square array with 5 colors.

Original entry on oeis.org

1, 52, 28564, 165770032, 10164078082036, 6584229526795818280, 45062665956031451017237456, 3258395057698765483724093981321824, 2489232886416012985921659124731697904597044, 20091032492258710696689787524926465967570325433558752

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..14

Main diagonal of A222144.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222144 = Cases[Import["https://oeis.org/A222144/b222144.txt", "Table"], {, }][[All, 2]];
a[n_] := A222144[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Sep 23 2019 *)

a(8)-a(10) from Alois P. Heinz, Apr 27 2012

cp's OEIS Frontend

A068239 1/2 the number of colorings of a 3 X 3 square array with n colors.

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A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

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A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

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A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

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A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

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A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

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A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

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A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

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