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A182793
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Number of n-colorings of the 8 X 8 X 8 triangular grid.
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12
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0, 0, 0, 6, 1031276544, 4826149802070660, 316827094291524894720, 1595091571660292411606250, 1592275064882420035249606656, 526249245643156296389047576104, 78022473527414400196098852126720, 6300701001267935948773824927446190
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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
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MAPLE
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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);
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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