A182795 Number of n-colorings of the 10 X 10 X 10 triangular grid.

0, 0, 0, 6, 6468240187392, 143635721907943000938060, 4861091521972177266672058368000, 2856800670438221106476061284736341250, 131028911804088893672445293407292154494976

Offset

0,4

Comments

The 10 X 10 X 10 triangular grid has 10 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 55 vertices and 135 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 (56, -1540, 27720, -367290, 3819816, -32468436, 231917400, -1420494075, 7575968400, -35607051480, 148902215280, -558383307300, 1889912732400, -5804731963800, 16253249498640, -41648951840265, 97997533741800, -212327989773900, 424655979547800, -785613562163430, 1346766106565880, -2142582442263900, 3167295784216200, -4355031703297275, 5574440580220512, -6646448384109072, 7384942649010080, -7648690600760440, 7384942649010080, -6646448384109072, 5574440580220512, -4355031703297275, 3167295784216200, -2142582442263900, 1346766106565880, -785613562163430, 424655979547800, -212327989773900, 97997533741800, -41648951840265, 16253249498640, -5804731963800, 1889912732400, -558383307300, 148902215280, -35607051480, 7575968400, -1420494075, 231917400, -32468436, 3819816, -367290, 27720, -1540, 56, -1).

Formula

a(n) = n^55 -135*n^54 + ... (see Maple program).

Maple

a:= n-> n^55 -135*n^54 +8964*n^53 -390222*n^52 +12525057*n^51 -316076903*n^50 +6530286070*n^49 -113573987769*n^48 +1696787220520*n^47 -22113112510550*n^46 +254428951045842*n^45 -2609511250718613*n^44 +24045856082285419*n^43 -200371113856491240*n^42 +1518133675627952270*n^41 -10506651071221868153*n^40 +66680463251797921915*n^39 -389373183471975572302*n^38 +2098028797385404193010*n^37
-10456871082871436486097*n^36 +48311408769374448761586*n^35 -207268123118278617037243*n^34 +827002152243388922174239*n^33 -3072694198727638003487979*n^32 +10641864949286796056022377*n^31 -34383949683339954923684782*n^30 +103704885062207595279156312*n^29 -292098504456226533053440510*n^28 +768501708532085822533190556*n^27 -1888698433570434475839725929*n^26 +4335279422341414825800378209*n^25
-9290907905051445440799000716*n^24 +18580084162229028469273798451*n^23 -34646102938311786771803477712*n^22 +60179271229381177090538625964*n^21 -97248893234106206859587981511*n^20 +145984266730291101055714541723*n^19 -203195282517216004808829603690*n^18 +261670683045031491886557091942*n^17 -310956138275834795608083550274*n^16 +339941943100528554861813262560*n^15
-340628682378318048979653175381*n^14 +311484260127833509262781795600*n^13 -258586709722348835998646850788*n^12 +193670730551369756737363762352*n^11 -129863868693889627423240097464*n^10 +77228998619164716149657770512*n^9 -40252487790410927197535447840*n^8 +18109784947870880558334595968*n^7 -6892748007729626216676319168*n^6 +2158618972888431826460898944*n^5 -534180587663008964293559296*n^4
+97953970795833012084624384*n^3 -11833494445627750018634752*n^2 +706434229524151535286272*n: seq(a(n), n=0..12);

Crossrefs

10th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182796.

Sequence in context: A058457 A182794 A072240 * A182796 A079683 A115528

Adjacent sequences: A182792 A182793 A182794 * A182796 A182797 A182798

Keyword

nonn,easy

Author

Alois P. Heinz, Dec 02 2010

Status

approved