A212767 Number of (w,x,y,z) with all terms in {0,...,n}, w even, x even, and w+x=y+z.

1, 1, 8, 10, 29, 35, 72, 84, 145, 165, 256, 286, 413, 455, 624, 680, 897, 969, 1240, 1330, 1661, 1771, 2168, 2300, 2769, 2925, 3472, 3654, 4285, 4495, 5216, 5456, 6273, 6545, 7464, 7770, 8797, 9139, 10280, 10660, 11921, 12341, 13728, 14190

Offset: 0

Author

Clark Kimberling, May 29 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

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

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && (Mod[x, 2] == 0 && w + x == y + z), s++], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]  (* A212767 *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 1, 8, 10, 29, 35, 72}, 50]

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

a(n) = (4*n^3+15*n^2+20*n+12+3*(n^2+4*n+4)*(-1)^n)/24. - Luce ETIENNE, Jun 03 2014