A212742 Number of (w,x,y,z) with all terms in {0,...,n} and max{w,x,y,z}<=2*min{w,x,y,z}.

1, 2, 17, 32, 97, 162, 337, 512, 881, 1250, 1921, 2592, 3697, 4802, 6497, 8192, 10657, 13122, 16561, 20000, 24641, 29282, 35377, 41472, 49297, 57122, 66977, 76832, 89041, 101250, 116161, 131072, 149057, 167042, 188497, 209952, 235297

Offset: 0

Author

Clark Kimberling, May 26 2012

Keywords

• nonn
• easy

Comments

Also, the number of (w,x,y,z) with all terms in {0,...,n} and the differences w-x, x-y, y-z all even.

a(n)+A212743(n) = n^4.

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[Max[w, x, y, z] <= 2 Min[w, x, y, z], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212742 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{1,2,17,32,97,162,337,512},40] (* Harvey P. Dale, May 14 2013 *)

Formula

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.: -(1+x^2)*(x^4+10*x^2+1) / ( (1+x)^3*(x-1)^5 ).

a(n) = A212740(n+1) for n>=0.