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

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

Offset: 1

Author

R. H. Hardin, Feb 24 2002

Keywords

• nonn

Links

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

Crossrefs

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

See A047938 for number of improper colorings.

Main diagonal of A078099.

Twice A207993 for n>1.

Programs

Mathematica

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 *)

Formula

For formula see A078099.

Extensions

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