A212561 Number of (w,x,y,z) with all terms in {1,...,n} and w + x = 2y + 2z.

0, 0, 1, 5, 12, 26, 45, 75, 112, 164, 225, 305, 396, 510, 637, 791, 960, 1160, 1377, 1629, 1900, 2210, 2541, 2915, 3312, 3756, 4225, 4745, 5292, 5894, 6525, 7215, 7936, 8720, 9537, 10421, 11340, 12330, 13357, 14459, 15600, 16820, 18081, 19425

Offset: 0

Author

Clark Kimberling, May 21 2012

Keywords

• nonn
• easy

Comments

Probably related to A199771 and A200252.

For a guide to related sequences, see A211795.

Except for the first term, a(n) is the number of undirected rook moves on an n X n chessboard, considered up to rotations but not reflections. - Hilko Koning, Aug 10 2025

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x == 2 y + 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212561 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,1,5,12,26},50] (* Harvey P. Dale, Dec 04 2016 *)
a[n_Integer?NonNegative] := ((n - 1) (2 n^2 + 1 - (-1)^n))/8
Table[a[n], {n, 0, 100}] (* Hilko Koning, Aug 10 2025 *)

PARI

concat([0,0], Vec(x^2*(x^3+x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100))) \\ Colin Barker, Feb 17 2015

Formula

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).

a(n) = (2*n^3-2*n^2+n-1-(n-1)*(-1)^n)/8 = (n-1)*(2*n^2+1-(-1)^n)/8. - Luce ETIENNE, Jul 26 2014

G.f.: x^2*(x^3+x^2+3*x+1) / ((x-1)^4*(x+1)^2). - Colin Barker, Feb 17 2015