A182796 Number of n-colorings of the 11 X 11 X 11 triangular grid.

0, 0, 0, 6, 894839431299072, 2669547726944484045356192220, 3453061562403499837458734621479403520, 32534816367748624110581496623513688165161250, 13865643738325095813931525301368809527451487174656, 719243085838104840090332816450418348485262159478161912

Offset: 0

Author

Alois P. Heinz, Dec 02 2010

Keywords

• nonn
• easy

Comments

The 11 X 11 X 11 triangular grid has 11 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 66 vertices and 165 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 (67, -2211, 47905, -766480, 9657648, -99795696, 869648208, -6522361560, 42757703560, -247994680648, 1285063345176, -5996962277488, 25371763481680, -97862516286480, 345780890878896, -1123787895356412, 3371363686069236, -9364899127970100, 24151581961607100, -57963796707857040, 129728497393775280, -271250494550621040, 530707489338171600, -972963730453314600, 1673497616379701112, -2703342303382594104, 4105075349580976232, -5864393356544251760, 7886597962249166160, -9989690752182277136, 11923179284862717872, -13413576695470557606, 14226520737620288370, -14226520737620288370, 13413576695470557606, -11923179284862717872, 9989690752182277136, -7886597962249166160, 5864393356544251760, -4105075349580976232, 2703342303382594104, -1673497616379701112, 972963730453314600, -530707489338171600, 271250494550621040, -129728497393775280, 57963796707857040, -24151581961607100, 9364899127970100, -3371363686069236, 1123787895356412, -345780890878896, 97862516286480, -25371763481680, 5996962277488, -1285063345176, 247994680648, -42757703560, 6522361560, -869648208, 99795696, -9657648, 766480, -47905, 2211, -67, 1).

Crossrefs

11th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795.

Programs

Maple

a:= n-> n^66 -165*n^65 +13430*n^64 -718830*n^63 [phone]*n^62 -888623847*n^61 [phone]*n^60 -493911980736*n^59 [phone]*n^58 -150915853835753*n^57 +2187810200892517*n^56 -28386731631190882*n^55 +332304034158619019*n^54 -3533226535570171926*n^53 +34313909582632869954*n^52 -305856530408381979601*n^51 +2512508789703297897295*n^50 -19089408783899171447224*n^49 +134562619568457264195163*n^48
-882441314560383975170374*n^47 +5396523102436821589146163*n^46 -30840476493483204890335403*n^45 +165009710808610594759616084*n^44 -827914124972290242846288614*n^43 +3900932089129512379033249682*n^42 -17282292209365903724659563631*n^41 +72070311947250436580694965993*n^40 -283166145176179540399078790292*n^39 +1049069241527084408399974095750*n^38 -3667220337345620153484655187124*n^37
+12102613021744672034697503592240*n^36 -37724138339405445177425698342523*n^35 +111095760575994820098618163390207*n^34 -309176068977052084408729303614893*n^33 +813185481965001199040935097964080*n^32 -2021374436814237148012243424806903*n^31 +4748186561462311698450896683155065*n^30 -10537422803434213322732080981201161*n^29 +22086052643134325938087794218181024*n^28
-43699620756746667796067005960087177*n^27 +81574844104346290652888156183655294*n^26 -143561350684851401447755384461673931*n^25 +237980280375008015726322556682052877*n^24 -371206816676060485457461990985198956*n^23 +544170012342342058668596490042636752*n^22 -748657464524219415245225971665770397*n^21 +965053026942268357862711436169935542*n^20 -1163371795450218690971885318270471694*n^19
+1308697520027710079307786302348771339*n^18 -1370319041971898252774123231153226918*n^17 +1331690339384350939067376866415236621*n^16 -1197068569703716329028295302490292938*n^15 +991428141596470240524919848774681738*n^14 -753054945934102362521837371999863872*n^13 +521731607147367465356546993487963024*n^12 -327563800253835254381288187488707872*n^11 +184908996556501805959894731292086336*n^10
-92949398227453879699243734196772032*n^9 +41108507052047410428558518243062272*n^8 -15751620136596962785464735723309056*n^7 +5123987337580699585298644858115072*n^6 -1376145015411556644420090237028352*n^5 +292997762191812894902503923634176*n^4 -46372215676408895763951507652608*n^3 +4850060647318928018465677025280*n^2 -251433237032021534887746912256*n:
seq(a(n), n=0..12);

Formula

a(n) = n^66 -165*n^65 + ... (see Maple program).