A212687 Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|

0, 1, 12, 67, 212, 527, 1096, 2045, 3496, 5621, 8580, 12591, 17852, 24627, 33152, 43737, 56656, 72265, 90876, 112891, 138660, 168631, 203192, 242837, 287992, 339197, 396916, 461735, 534156, 614811, 704240, 803121, 912032, 1031697

Offset: 0

Author

Clark Kimberling, May 25 2012

Keywords

• nonn
• easy

Comments

a(n)+A212688(n)=n^4.

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

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 Abs[w - x] < n + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212687 *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 1, 12, 67, 212, 527, 1096}, 40]

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 + 9*x^2 + 32*x^3 + 28*x^4 + 13*x^5 + x^6)/(1 - 3*x + x^2 + 5*x^3 - 5*x^4 - x^5 + 3*x^6 - x^7).