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A159299 Number of n-colorings of the 4 X 4 Sudoku graph.

%I A159299 #32 Jul 06 2025 18:46:49
%S A159299 0,0,0,0,288,166560,33539040,2350746720,75756999360,1388552614848,
%T A159299 16744788486720,146769785743680,1002373493948640,5606534724167520,
%U A159299 26640793339768608,110556058012152480,409297168707073920,1374572399886053760,4243833928227876480
%N A159299 Number of n-colorings of the 4 X 4 Sudoku graph.
%C A159299 The 4 X 4 Sudoku graph is a septic graph on 16 vertices and 56 edges. a(n) gives the number of 4 X 4 Sudoku solutions, if each of up to n numbers is allowed only once in every row, column and block.
%H A159299 Alois P. Heinz, <a href="/A159299/b159299.txt">Table of n, a(n) for n = 0..10000</a>
%H A159299 Timme, Marc; van Bussel, Frank; Fliegner, Denny; Stolzenberg, Sebastian (2009) "Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions", New J. Phys. 11 023001, doi: <a href="http://dx.doi.org/10.1088/1367-2630/11/2/023001">10.1088/1367-2630/11/2/023001</a>.
%H A159299 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A159299 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mathematics_of_Sudoku">Mathematics of Sudoku</a>
%H A159299 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sudoku">Sudoku</a>
%H A159299 Wikipedia, <a href="https://en.wikipedia.org/wiki/Algorithmics_of_sudoku">Sudoku algorithms</a>
%H A159299 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680, -2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380, -680,136,-17,1).
%F A159299 a(n) = n^16 -56*n^15 + ... (see Maple program).
%F A159299 G.f.: -96*x^4*(343316843*x^12 +4128584684*x^11 +20203233398*x^10 +50370257700*x^9 +68017469565*x^8 +50271571704*x^7 +20027437332*x^6 +4145554824*x^5 +419198325*x^4 +18781660*x^3 +320278*x^2 +1684*x +3)/ (x-1)^17. - _Colin Barker_, Aug 04 2012
%e A159299 For n=4 colors one of the 288 possible colorings is given by this Sudoku:
%e A159299   +---+---+
%e A159299   |1 2|3 4|
%e A159299   |4 3|2 1|
%e A159299   +---+---+
%e A159299   |3 1|4 2|
%e A159299   |2 4|1 3|
%e A159299   +---+---+ .
%p A159299 a:= n-> n^16 -56*n^15 +1492*n^14 -25072*n^13 +296918*n^12 -2621552*n^11 +17795572*n^10 -94352168*n^9 +392779169*n^8 -1279118840*n^7 +3217758336*n^6 -6107865464*n^5 +8413745644*n^4 -7877463064*n^3 +4436831332*n^2 -1117762248*n: seq(a(n), n=0..20);
%Y A159299 Cf. A107739, A182866.
%K A159299 nonn,easy
%O A159299 0,5
%A A159299 _Alois P. Heinz_, Apr 09 2009