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

0,4

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).

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

Maple

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);

Prog

(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

Crossrefs

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

Sequence in context: A363500 A193150 A307896 * A058431 A045518 A058457

Adjacent sequences: A182790 A182791 A182792 * A182794 A182795 A182796

Keyword

nonn,easy

Author

Alois P. Heinz, Dec 02 2010

Status

approved