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, 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.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4,12).

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

Adjacent sequences: A147831 A147832 A147833 * A147835 A147836 A147837

Keyword

nonn,easy

Author

Roger L. Bagula, Nov 14 2008

Status

approved