A205492 Expansion of (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)).

1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976, 87369, 203915, 471546, 1082849, 2473535, 5627684, 12765052, 28887838, 65260270, 147233926, 331842395, 747355066, 1682185342, 3784718431, 8512408455, 19141037360, 43032743620

Offset

0,2

Comments

See array A205497 regarding association of this sequence with generating functions for the rows of the array form of A050446.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

L. E. Jeffery, Unit-primitive matrices

Index entries for linear recurrences with constant coefficients, signature (7,-17,12,15,-26,3,13,-5,-2,1).

Formula

a(n) = 7*a(n-1) - 17*a(n-2) + 12*a(n-3) + 15*a(n-4) - 26*a(n-5) + 3*a(n-6) + 13*a(n-7) - 5*a(n-8) - 2*a(n-9) + a(n-10), n>9, {a(m)} = {1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976}, m=0,...,9.

CONJECTURE 1. a(n) = M_{n,2} = M_{2,n}, where M = A205497.

CONJECTURE 2. lim_{n -> infinity) a(n+1)/a(n) = (2*cos(Pi/7))^2-1 = A116425-1 = spectral radius of the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,2} = [0,0,1; 0,1,1; 1,1,1].

Maple

seq(coeff(series((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 04 2020

Mathematica

LinearRecurrence[{7, -17, 12, 15, -26, 3, 13, -5, -2, 1}, {1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976}, 30] (* Harvey P. Dale, Mar 26 2013 *)
CoefficientList[Series[(1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), {x, 0, 30}], x] (* G. C. Greubel, Jan 04 2020 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3))) \\ G. C. Greubel, Jan 04 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)) )); // G. C. Greubel, Jan 04 2020
(Sage)
def A205492_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)) ).list()
A205492_list(30) # G. C. Greubel, Jan 04 2020

Crossrefs

Cf. A050446, A050447, A205497.

Sequence in context: A119359 A055366 A160607 * A109756 A055580 A364635

Adjacent sequences: A205489 A205490 A205491 * A205493 A205494 A205495

Keyword

nonn

Author

L. Edson Jeffery, Jan 28 2012

Status

approved