A212893 Number of quadruples (w,x,y,z) with all terms in {0,...,n} such that w-x, x-y, and y-z all have the same parity.

1, 4, 25, 64, 169, 324, 625, 1024, 1681, 2500, 3721, 5184, 7225, 9604, 12769, 16384, 21025, 26244, 32761, 40000, 48841, 58564, 70225, 82944, 97969, 114244, 133225, 153664, 177241, 202500, 231361, 262144, 297025, 334084, 375769

Offset: 0

Author

Clark Kimberling, May 30 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Sum of odd integers between 1 and (n+1)^2. - Réjean Labrie, Jan 14 2014

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A000982, A211795.

Programs

Maple

A212893 := n->ceil((n+1)^2/2)^2; seq(A212893(k), k=1..100); # Wesley Ivan Hurt, Jun 14 2013

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Mod[w - x, 2] == Mod[x - y, 2] == Mod[y - z, 2], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* this sequence *)
Sqrt[m]  (* A000982 except for offset *)

Formula

a(n) = (A000982(n+1))^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.: f(x)/g(x), where f(x) = -1 - 2*x - 15*x^2 - 12*x^3 - 15*x^4 - 2*x^5 - x^6 and g(x) = ((-1+x)^5)*(1+x)^3.