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

%I A182795 #18 Jan 21 2024 11:57:07
%S A182795 0,0,0,6,6468240187392,143635721907943000938060,
%T A182795 4861091521972177266672058368000,
%U A182795 2856800670438221106476061284736341250,131028911804088893672445293407292154494976
%N A182795 Number of n-colorings of the 10 X 10 X 10 triangular grid.
%C A182795 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.
%H A182795 Alois P. Heinz, <a href="/A182795/b182795.txt">Table of n, a(n) for n = 0..1000</a>
%H A182795 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182795 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182795 <a href="/index/Rec#order_56">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A182795 a(n) = n^55 -135*n^54 + ... (see Maple program).
%p A182795 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
%p A182795 -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
%p A182795 -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
%p A182795 -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
%p A182795 +97953970795833012084624384*n^3 -11833494445627750018634752*n^2 +706434229524151535286272*n: seq(a(n), n=0..12);
%Y A182795 10th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182796.
%K A182795 nonn,easy
%O A182795 0,4
%A A182795 _Alois P. Heinz_, Dec 02 2010