A212508 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<3z.

0, 1, 12, 56, 168, 418, 837, 1554, 2640, 4209, 6375, 9373, 13176, 18161, 24402, 32110, 41472, 52948, 66339, 82384, 101100, 122801, 147741, 176665, 209088, 246225, 287976, 334764, 386904, 445486, 509625, 581126, 659712, 745921

Offset: 0

Author

Clark Kimberling, May 19 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, -1, -4, 0, 2, 0, -2, 0, 4, 1, -2, -2, 0, 1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w < 2 x && y < 3 z, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]]; Map[t[#] &, Range[0, 50]]   (* A212508 *)
Table[n^2/24 + n^3/3 + 5*n^4/8 - 1/12*Floor[n/6] - 1/4*n^2*Floor[n/3] - (n/12 + 5*n^2/12) * Floor[n/2] + 1/12*Floor[(1 + n)/6] + 1/4*n^2*Floor[(1 + n)/3], {n, 0, 50}] (* Vaclav Kotesovec, Dec 11 2015 *)

Formula

a(n)=2a(n-2)+2a(n-3)-a(n-4)-4a(n-5)+2a(n-7)-2a(n-9)+4a(n-11)+a(n-12)-2a(n-13)-2a(n-14)+a(n-16).

G.f.: -x*((1 + 12*x + 54*x^2 + 142*x^3 + 283*x^4 + 405*x^5 + 486*x^6 + 520*x^7 + 493*x^8 + 386*x^9 + 265*x^10 + 136*x^11 + 47*x^12 + 9*x^13 + x^14)/((-1 + x)^5*(1 + x)^3 * (1 - x + x^2)*(1 + x + x^2)^3)). - Vaclav Kotesovec, Dec 11 2015