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Coefficient expansion of the characteristic polynomial of the {3,4,5} simplex matrix: M = {{0, 3, 0}, {0, 0, 4}, {1, 1, 1}}; p(x)=12 + 4 x + x^2 - x^3.
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%I #9 Jul 16 2023 09:38:38

%S 1,1,5,21,53,197,661,2085,7093,23365,76757,255333,842741,2785157,

%T 9220117,30473637,100775989,333311941,1102099541,3644659173,

%U 12052800629,39856631813,131803744405,435863879205,1441358438581,4766458888261,15762259193045,52124396009061

%N Coefficient expansion of the characteristic polynomial of the {3,4,5} simplex matrix: M = {{0, 3, 0}, {0, 0, 4}, {1, 1, 1}}; p(x)=12 + 4 x + x^2 - x^3.

%C The {3,4,5} represents one of the few integer based triangular tilings of the plane by triangles.

%H Colin Barker, <a href="/A147834/b147834.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,12).

%F p(x)=12 + 4 x + x^2 - x^3; a(n)=coefficient_expansion(-x^3*p(1/x)).

%F From _Colin Barker_, Jan 05 2018: (Start)

%F G.f.: 1 / (1 - x - 4*x^2 - 12*x^3).

%F a(n) = a(n-1) + 4*a(n-2) + 12*a(n-3) for n>2.

%F (End)

%t f[x_] = 12 + 4 x + x^2 - x^3; g[x] = ExpandAll[ -x^3*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]

%t LinearRecurrence[{1,4,12},{1,1,5},30] (* _Harvey P. Dale_, Jul 16 2023 *)

%o (PARI) Vec(1 / (1 - x - 4*x^2 - 12*x^3) + O(x^40)) \\ _Colin Barker_, Jan 05 2018

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Nov 14 2008