A212748 Number of (w,x,y,z) with all terms in {0,...,n} and w=2*floor((x+y+z)/2).

1, 4, 17, 20, 53, 56, 117, 120, 217, 220, 361, 364, 557, 560, 813, 816, 1137, 1140, 1537, 1540, 2021, 2024, 2597, 2600, 3273, 3276, 4057, 4060, 4957, 4960, 5981, 5984, 7137, 7140, 8433, 8436, 9877, 9880, 11477, 11480, 13241, 13244, 15177

Offset

0,2

Comments

For a guide to related sequences, see A211795.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).

G.f.: (1+3*x+10*x^2-6*x^3-3*x^4+3*x^5) / ((1+x)^3*(x-1)^4).

From Colin Barker, Jan 29 2016: (Start)

a(n) = (2*n^3+3*((-1)^n+5)*n^2+(15*(-1)^n+37)*n+12)/12.

a(n) = (n^3+9*n^2+26*n+6)/6 for n even.

a(n) = (n^3+6*n^2+11*n+6)/6 for n odd. (End)

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w == 2*Floor[(x + y + z)/2], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 45]]   (* A212748 *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 4, 17, 20, 53, 56, 117}, 50] (* Harvey P. Dale, Jun 08 2018 *)

Prog

(PARI) Vec((1+3*x+10*x^2-6*x^3-3*x^4+3*x^5)/((1+x)^3*(x-1)^4) + O(x^100)) \\ Colin Barker, Jan 29 2016

Crossrefs

Cf. A211795.

Sequence in context: A031444 A031033 A128981 * A032828 A193379 A022134

Adjacent sequences: A212745 A212746 A212747 * A212749 A212750 A212751

Keyword

nonn,easy

Author

Clark Kimberling, May 28 2012

Status

approved