A212503 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<2z.

0, 1, 9, 49, 144, 361, 729, 1369, 2304, 3721, 5625, 8281, 11664, 16129, 21609, 28561, 36864, 47089, 59049, 73441, 90000, 109561, 131769, 157609, 186624, 219961, 257049, 299209, 345744, 398161, 455625, 519841, 589824, 667489, 751689

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 (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A211795.

Programs

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]]   (* A212503 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,9,49,144,361,729,1369},40] (* Harvey P. Dale, Jun 14 2017 *)

Formula

a(n) = (A077043(n))^2.

a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8).

G.f.: x*(1+x+x^2)*(1+6*x+22*x^2+6*x^3+x^4)/((1+x)^3*(1-x)^5). [Bruno Berselli, May 31 2012]