A091940 Given n colors, sequence gives number of ways to color the vertices of a square such that no edge has the same color on both of its vertices.

0, 2, 18, 84, 260, 630, 1302, 2408, 4104, 6570, 10010, 14652, 20748, 28574, 38430, 50640, 65552, 83538, 104994, 130340, 160020, 194502, 234278, 279864, 331800, 390650, 457002, 531468, 614684, 707310, 810030, 923552, 1048608, 1185954, 1336370, 1500660

Offset

1,2

Comments

Also equals the number of pairs of pairs ((a_1,a_2),(b_1,b_2)) that are disjoint (a_i != b_j) where all elements belong to {1,...,n}. See A212085. - Lewis Baxter, Mar 06 2023

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

OEIS Wiki, Colorings of square grid graphs

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2*C(n,2) + 12*C(n,3) + 24*C(n,4) = n*(n-1)*(n^2-3*n+3).

a(n) = (n-1) + (n-1)^4. - Rainer Rosenthal, Dec 03 2006

G.f.: 2*x^2*(1+4*x+7*x^2)/(1-x)^5. a(n) = 2*A027441(n-1). - R. J. Mathar, Sep 09 2008

For n > 1, a(n) = floor(n^7/(n^3-1)). - Gary Detlefs, Feb 10 2010

a(n) = 2 * A000217(n-1) * A002061(n-1), n >= 1. - Daniel Forgues, Jul 14 2016

E.g.f.: exp(x)*x^2*(1 + x)^2. - Stefano Spezia, Oct 08 2022

Example

a(4) = 84 since there are 84 different ways to color the vertices of a square with 4 colors such that no two vertices that share an edge are the same color.
There are 4 possible colors for the first vertex and 3 for the second vertex. For the third vertex, divide into two cases: the third vertex can be the same color as the first vertex, and then the fourth vertex has 3 possible colors (4 * 3 * 1 * 3 = 36 colorings). Or the third vertex can be a different color from the first vertex, and then the fourth vertex has 2 possible colors (4 * 3 * 2 * 2 = 48 colorings). So there are a total of 36 + 48 = 84. - Michael B. Porter, Jul 24 2016

Maple

a := n -> (n-1)+(n-1)^4; for n to 35 do a(n) end do; # Rainer Rosenthal, Dec 03 2006

Mathematica

Table[2Binomial[n, 2] + 12Binomial[n, 3] + 24Binomial[n, 4], {n, 35}] (* Robert G. Wilson v, Mar 16 2004 *)
Table[(n-1)^4+(n-1), {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)

Crossrefs

Column k=2 of A212085.

Cf. A027441, A182368, A182406.

Cf. A000217, A002061.

Sequence in context: A323947 A064057 A176496 * A068605 A343514 A360301

Adjacent sequences: A091937 A091938 A091939 * A091941 A091942 A091943

Keyword

nonn,easy

Author

Ryan Witko (witko(AT)nyu.edu), Mar 11 2004

Extensions

More terms from Robert G. Wilson v, Mar 16 2004

Status

approved