A213396 Number of (w,x,y) with all terms in {0,...,n} and 2*w < |x+y-w|.

0, 3, 9, 21, 42, 72, 114, 171, 243, 333, 444, 576, 732, 915, 1125, 1365, 1638, 1944, 2286, 2667, 3087, 3549, 4056, 4608, 5208, 5859, 6561, 7317, 8130, 9000, 9930, 10923, 11979, 13101, 14292, 15552, 16884, 18291, 19773, 21333, 22974, 24696

Offset: 0

Author

Clark Kimberling, Jun 12 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Also, integer values of (m^3+1)/3 for m>0. - Bruno Berselli, Jan 19 2013

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A060999, A212959, A213397.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[2 w < Abs[x + y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]]; m = Map[t[#] &, Range[0, 60]]
CoefficientList[ Series[(3 (x + x^3))/((-1 + x)^4 (1 + x + x^2)), {x, 0, 41}], x] (* or *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 3, 9, 21, 42, 72}, 41] (* Robert G. Wilson v, Dec 22 2017 *)

PARI

x='x+O('x^99); concat([0], Vec(3*x*(1+x^2)/((1-x)^4*(1+x+x^2)))) \\ Altug Alkan, Dec 22 2017

Formula

a(n) = (n+2)*(n+1)*n/3 + floor((n-1)/3) + 1.

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).

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

a(n) + A213397(n) = (n+1)^3.

a(n) = 3*A060999(n). - Bruno Berselli, Dec 22 2017

Extensions

Corrected the title. Robert G. Wilson v, Dec 22 2017