A212563 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=2y+2z.

0, 1, 16, 78, 240, 577, 1182, 2172, 3680, 5865, 8900, 12986, 18336, 25193, 33810, 44472, 57472, 73137, 91800, 113830, 139600, 169521, 204006, 243508, 288480, 339417, 396812, 461202, 533120, 613145, 701850, 799856, 907776, 1026273, 1156000, 1297662, 1451952

Offset

0,3

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 (3,-1,-5,5,1,-3,1).

Formula

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

From Colin Barker, Dec 10 2015: (Start)

a(n) = 1/96*(82*n^4+36*n^3-16*n^2-6*((-1)^n-1)*n+9*((-1)^n-1)).

G.f.: x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2).

(End)

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x <= 2 y + 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212563 *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 1, 16, 78, 240, 577, 1182}, 40] (* Harvey P. Dale, Aug 28 2020 *)

Prog

(PARI) concat(0, Vec(x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2) + O(x^50))) \\ Colin Barker, Dec 10 2015

Crossrefs

Cf. A211795.

Sequence in context: A302412 A303179 A061995 * A210324 A250231 A212896

Adjacent sequences: A212560 A212561 A212562 * A212564 A212565 A212566

Keyword

nonn,easy

Author

Clark Kimberling, May 21 2012

Status

approved