

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.


25



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
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OFFSET

1,2


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*(n1)*(n^23*n+3).
a(n) = (n1) + (n1)^4.  Rainer Rosenthal, Dec 03 2006
G.f.: 2*x^2*(1+4*x+7*x^2)/(1x)^5. a(n) = 2*A027441(n1).  R. J. Mathar, Sep 09 2008
For n > 1, a(n) = floor(n^7/(n^31)).  Gary Detlefs, Feb 10 2010
a(n) = 2 * A000217(n1) * A002061(n1), n >= 1.  Daniel Forgues, Jul 14 2016


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 > (n1)+(n1)^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[(n1)^4+(n1), {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)


CROSSREFS

Cf. A027441, A182368, A182406.
Sequence in context: A323947 A064057 A176496 * A068605 A343514 A070171
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



