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A182793 Number of n-colorings of the 8 X 8 X 8 triangular grid.

%I A182793 #24 Jan 21 2024 11:54:57
%S A182793 0,0,0,6,1031276544,4826149802070660,316827094291524894720,
%T A182793 1595091571660292411606250,1592275064882420035249606656,
%U A182793 526249245643156296389047576104,78022473527414400196098852126720,6300701001267935948773824927446190
%N A182793 Number of n-colorings of the 8 X 8 X 8 triangular grid.
%C A182793 The 8 X 8 X 8 triangular grid has 8 rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has 36 vertices and 84 edges altogether.
%H A182793 Alois P. Heinz, <a href="/A182793/b182793.txt">Table of n, a(n) for n = 0..1000</a>
%H A182793 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182793 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182793 <a href="/index/Rec#order_37">Index entries for linear recurrences with constant coefficients</a>, signature (37, -666, 7770, -66045, 435897, -2324784, 10295472, -38608020, 124403620, -348330136, 854992152, -1852482996, 3562467300, -6107086800, 9364199760, -12875774670, 15905368710, -17672631900, 17672631900, -15905368710, 12875774670, -9364199760, 6107086800, -3562467300, 1852482996, -854992152, 348330136, -124403620, 38608020, -10295472, 2324784, -435897, 66045, -7770, 666, -37, 1).
%F A182793 a(n) = n^36 -84*n^35 + ... (see Maple program).
%F A182793 a(n) = (n^30 + ... )*n*(n-1)*(n-2)^4 (see PARI program), therefore all terms are divisible by 6. - _M. F. Hasler_, Dec 02 2010
%p A182793 a:= n-> n^36 -84*n^35 +3437*n^34 -91266*n^33 +1767948*n^32 -26626641*n^31 +324474230*n^30 -3287527515*n^29 +28241112564*n^28 -208720581316*n^27 +1342098781876*n^26 -7574085510428*n^25 +37773151152128*n^24 -167375021582772*n^23 +661739022592885*n^22 -2341944556478962*n^21 +7436934470326959*n^20 -21224613967949058*n^19 +54488667645973816*n^18 -125859887740997948*n^17 +261444368727996373*n^16 -487829426279117443*n^15 +816027319948726718*n^14 -1220298815193350831*n^13 +1625157969312740380*n^12 -1917859440184087949*n^11 +1992559474100473934*n^10 -1807335902805940076*n^9 +1415695106519940144*n^8 -943996557462968752*n^7 +525570615466126368*n^6 -237792323595423264*n^5 +84014216771282688*n^4 -21747100909979904*n^3 +3668087119290368*n^2 -302469084548608*n: seq(a(n), n=0..12);
%o A182793 (PARI) a(n) = n*(n-1)*(n-2)^4*(n^30 -15*(5*n^20 -182*n^19 -73212*n^17 +968723*n^16 -10321679*n^15 +90965902*n^14 -42239514291692*n^5 +728948069669224)*n^9 -64240*n^27 +10138842074*n^22 -64422107890*n^21 +353781404418*n^20 -1692797609642*n^19 +7100833446102*n^18 -26231755759998*n^17 +85617623199383*n^16 -247408302649363*n^15 -1437889343008038*n^13 +2888477744794634*n^12 -5124456558208194*n^11 +8000185529836163*n^10 +12990665090694358*n^8 -13287807554341505*n^7 +11549829535832291*n^6 -8378308904565234*n^5 +4943464695686292*n^4 -2282977532565696*n^3 +775401219820384*n^2 -172542491602784*n +18904317784288) \\ - _M. F. Hasler_, Dec 02 2010
%Y A182793 8th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182794, A182795, A182796.
%K A182793 nonn,easy
%O A182793 0,4
%A A182793 _Alois P. Heinz_, Dec 02 2010