OFFSET
1,2
COMMENTS
Column 2 of A188277.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000 (first 200 terms from R. H. Hardin)
Robert Israel, Maple-assisted proof of empirical g.f.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1,0,0,0,0,1,-2,0,2,-1).
FORMULA
Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-9) - 2*a(n-10) + 2*a(n-12) - a(n-13).
Empirical g.f.: x*(1 + x - 3*x^2 + 2*x^3 - x^4 + x^5 - x^6 + x^7 + x^8 - 2*x^9 + 2*x^11 - x^12) / ((1 - x)^4*(1 + x)*(1 + x + x^2)*(1 + x^3 + x^6)). - Colin Barker, Apr 27 2018
From Robert Israel, Feb 07 2019: (Start)
Empirical g.f. verified: see link.
a(n) - (n^3/108 + (5/72)*n^2 + (19/36)*n) is periodic with period 18. (End)
EXAMPLE
All solutions for n=8 k=2:
.-1...-2...-2....0...-2...-2...-1...-2...-2...-2...-1...-2...-2...-1...-2
..0....0...-2....0...-1...-1...-1...-2...-2...-2...-1...-2...-1...-1...-2
..0....0...-2....0....0...-1...-1...-2....0...-2....0...-1...-1...-1...-1
..0....0...-1....0....0....0...-1...-2....0....0....0...-1...-1....0....0
..0....0....1....0....0....0....1....2....0....0....0....1....1....0....0
..0....0....2....0....0....1....1....2....0....2....0....1....1....1....1
..0....0....2....0....1....1....1....2....2....2....1....2....1....1....2
..1....2....2....0....2....2....1....2....2....2....1....2....2....1....2
MAPLE
S:= series((y^8 - y^7 + y^6 - y^5 + y^4 - y^3 + y^2 - y + 1)/(
y^13 - 2*y^12 + 2*y^10 - y^9 - y^4 + 2*y^3 - 2*y + 1), y, 101):
seq(coeff(S, y, n), n=1..100); # Robert Israel, Feb 07 2019
MATHEMATICA
f[z_] := (z^2 - z + 1)*(z^6 - z^3 + 1)/((z - 1)^4*(z + 1)*(z^2 + z + 1)*(z^6 + z^3 + 1)); CoefficientList[Series[f[z], {z, 0, 100}], z] (* By the result of Robert Israel, Peter Luschny, Feb 07 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Mar 26 2011
STATUS
approved