A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.

0, 1, 12, 56, 168, 399, 810, 1480, 2496, 3965, 6000, 8736, 12312, 16891, 22638, 29744, 38400, 48825, 61236, 75880, 93000, 112871, 135762, 161976, 191808, 225589, 263640, 306320, 353976, 406995, 465750, 530656, 602112, 680561, 766428

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

Crossrefs

Cf. A211795.

Programs

Magma

[(2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32: n in [0..34]]; // Bruno Berselli, May 31 2012

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y <= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212507 *)
CoefficientList[Series[x (1 + 2 x) (1 + 7 x + 7 x^2 + 3 x^3)/((1 + x)^2 (1 - x)^5), {x, 0, 34}], x] (* Bruno Berselli, May 31 2012 *)

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).

G.f.: x*(1+2*x)*(1+7*x+7*x^2+3*x^3)/((1+x)^2*(1-x)^5). [Bruno Berselli, May 31 2012]

a(n) = (2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32. [Bruno Berselli, May 31 2012]

a(n) = A006578(n) * A077043(n). - Alois P. Heinz, May 31 2012