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Number of n-colorings of the 11 X 11 X 11 triangular grid.
12

%I #22 Jan 21 2024 11:58:31

%S 0,0,0,6,894839431299072,2669547726944484045356192220,

%T 3453061562403499837458734621479403520,

%U 32534816367748624110581496623513688165161250,13865643738325095813931525301368809527451487174656,719243085838104840090332816450418348485262159478161912

%N Number of n-colorings of the 11 X 11 X 11 triangular grid.

%C 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.

%H Alois P. Heinz, <a href="/A182796/b182796.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>

%H <a href="/index/Rec#order_67">Index entries for linear recurrences with constant coefficients</a>, 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).

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

%p a:= n-> n^66 -165*n^65 +13430*n^64 -718830*n^63 +28457415*n^62 -888623847*n^61 +22794225600*n^60 -493911980736*n^59 +9226616834936*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

%p -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

%p +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

%p -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

%p +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

%p -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:

%p seq(a(n), n=0..12);

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

%K nonn,easy

%O 0,4

%A _Alois P. Heinz_, Dec 02 2010