A068245 - OEIS

%I #18 Sep 08 2022 08:45:05

%S 1,7616,5141250,552093440,20631905875,395001645696,4771909547076,

%T 41190314035200,275192443300005,1502690499112000,6971521964029766,

%U 28275884687022336,102456840191225975,337289521082456320,1022310183284613000,2883605488481550336,7636012822945480521

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

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

%H Alois P. Heinz, <a href="/A068245/b068245.txt">Table of n, a(n) for n = 3..1000</a>

%F From _Alois P. Heinz_, May 02 2012: (Start)

%F 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.

%F 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)

%p 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:

%p seq(a(n), n=3..40); #  _Alois P. Heinz_, May 02 2012

%o (Magma) [(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

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

%K nonn,easy

%O 3,2

%A _R. H. Hardin_, Feb 24 2002

%E Extended beyond a(15) by _Alois P. Heinz_, May 02 2012