OFFSET
0,2
COMMENTS
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->oo} 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
KEYWORD
nonn,changed
AUTHOR
L. Edson Jeffery, Jan 28 2012
STATUS
approved