A188270 Number of nondecreasing strings of numbers (x(i), i=1..n) in -2..2 with sum x(i)^3 equal to 0.

1, 3, 3, 6, 6, 10, 10, 15, 17, 23, 27, 34, 40, 48, 56, 65, 75, 87, 99, 114, 128, 146, 162, 183, 201, 225, 247, 274, 300, 330, 360, 393, 427, 463, 501, 542, 584, 630, 676, 727, 777, 833, 887, 948, 1008, 1074, 1140, 1211, 1283, 1359, 1437, 1518, 1602, 1690, 1780, 1875

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

Cf. A188277.

Sequence in context: A182843 A358558 A008805 * A026925 A343481 A237665

Adjacent sequences: A188267 A188268 A188269 * A188271 A188272 A188273

Keyword

nonn,easy

Author

R. H. Hardin, Mar 26 2011

Status

approved