A211440 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w+3x+3y=0.

1, 3, 5, 17, 23, 29, 53, 63, 73, 109, 123, 137, 185, 203, 221, 281, 303, 325, 397, 423, 449, 533, 563, 593, 689, 723, 757, 865, 903, 941, 1061, 1103, 1145, 1277, 1323, 1369, 1513, 1563, 1613, 1769, 1823, 1877, 2045, 2103, 2161, 2341, 2403, 2465

Offset: 0

Author

Clark Kimberling, Apr 11 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A211422.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1)

Crossrefs

Cf. A211422.

Programs

Maple

seq(op([10*j^2+6*j+1, 10*j^2 + 10*j + 3, 10*j^2 + 14*j + 5]),j=0..30); # Robert Israel, Apr 03 2019

Mathematica

t[n_] := t[n] = Flatten[Table[2 w + 3 x + 3 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 30}]  (* A211440 *)
(t - 1)/2                    (* integers *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,5,17,23,29,53},60] (* Harvey P. Dale, Aug 29 2021 *)

Formula

Conjectures from Colin Barker, May 15 2017: (Start)

G.f.: (1 + 3*x + x^2)*(1 - x + 4*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2).

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6.

(End)

From Robert Israel, Apr 03 2019: (Start)

a(3*j) = 10*j^2+6*j+1.

a(3*j+1) = 10*j^2 + 10*j + 3.

a(3*j+2) = 10*j^2 + 14*j + 5.

This has the conjectured g.f. and recurrence. (End)