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

%I A182796 #22 Jan 21 2024 11:58:31
%S A182796 0,0,0,6,894839431299072,2669547726944484045356192220,
%T A182796 3453061562403499837458734621479403520,
%U A182796 32534816367748624110581496623513688165161250,13865643738325095813931525301368809527451487174656,719243085838104840090332816450418348485262159478161912
%N A182796 Number of n-colorings of the 11 X 11 X 11 triangular grid.
%C A182796 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 A182796 Alois P. Heinz, <a href="/A182796/b182796.txt">Table of n, a(n) for n = 0..1000</a>
%H A182796 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182796 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182796 <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 A182796 a(n) = n^66 -165*n^65 + ... (see Maple program).
%p A182796 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 A182796 -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 A182796 +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 A182796 -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 A182796 +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 A182796 -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 A182796 seq(a(n), n=0..12);
%Y A182796 11th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795.
%K A182796 nonn,easy
%O A182796 0,4
%A A182796 _Alois P. Heinz_, Dec 02 2010