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

0, 0, 0, 6, 1031276544, 4826149802070660, 316827094291524894720, 1595091571660292411606250, 1592275064882420035249606656, 526249245643156296389047576104, 78022473527414400196098852126720, 6300701001267935948773824927446190

Offset: 0

Author

Alois P. Heinz, Dec 02 2010

Keywords

• nonn
• easy

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Chromatic polynomial
• Wikipedia, Triangular grid graph
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

8th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182794, A182795, A182796.

Programs

Maple

a:= n-> n^36 -84*n^35 +3437*n^34 -91266*n^33 +1767948*n^32 -26626641*n^31 [phone]*n^30 -3287527515*n^29 [phone]*n^28 -208720581316*n^27 [phone]*n^26 -7574085510428*n^25 [phone]*n^24 -167375021582772*n^23 [phone]*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);

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 [phone]*n^14 -42239514291692*n^5 [phone])*n^9 -64240*n^27 [phone]*n^22 -64422107890*n^21 [phone]*n^20 -1692797609642*n^19 [phone]*n^18 -26231755759998*n^17 [phone]*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 [phone]*n^2 -172542491602784*n [phone]) \\ - M. F. Hasler, Dec 02 2010

Formula

a(n) = n^36 -84*n^35 + ... (see Maple program).

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