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A147834
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.
1
1, 1, 5, 21, 53, 197, 661, 2085, 7093, 23365, 76757, 255333, 842741, 2785157, 9220117, 30473637, 100775989, 333311941, 1102099541, 3644659173, 12052800629, 39856631813, 131803744405, 435863879205, 1441358438581, 4766458888261, 15762259193045, 52124396009061
OFFSET
0,3
COMMENTS
The {3,4,5} represents one of the few integer based triangular tilings of the plane by triangles.
FORMULA
p(x)=12 + 4 x + x^2 - x^3; a(n)=coefficient_expansion(-x^3*p(1/x)).
From Colin Barker, Jan 05 2018: (Start)
G.f.: 1 / (1 - x - 4*x^2 - 12*x^3).
a(n) = a(n-1) + 4*a(n-2) + 12*a(n-3) for n>2.
(End)
MATHEMATICA
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}]
LinearRecurrence[{1, 4, 12}, {1, 1, 5}, 30] (* Harvey P. Dale, Jul 16 2023 *)
PROG
(PARI) Vec(1 / (1 - x - 4*x^2 - 12*x^3) + O(x^40)) \\ Colin Barker, Jan 05 2018
CROSSREFS
Sequence in context: A272013 A219219 A272810 * A160378 A201440 A096942
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Nov 14 2008
STATUS
approved