A202253 Number of zero-sum -n..n arrays of 3 elements with adjacent element differences also in -n..n.

3, 9, 17, 27, 41, 57, 75, 97, 121, 147, 177, 209, 243, 281, 321, 363, 409, 457, 507, 561, 617, 675, 737, 801, 867, 937, 1009, 1083, 1161, 1241, 1323, 1409, 1497, 1587, 1681, 1777, 1875, 1977, 2081, 2187, 2297, 2409, 2523, 2641, 2761, 2883, 3009, 3137, 3267

Offset: 1

Author

R. H. Hardin, Dec 14 2011

Keywords

• nonn

Comments

Row 3 of A202252.

It appears that A202253 is also the number of ordered triples (w,x,y) with all terms in {-n,...,n} such that w+2x+3y=0; see the Mathematica and Example sections. - Clark Kimberling, Apr 10 2012

Examples

			Some solutions for n=10:
   7   9   6   4  -2   3  -3  -8   3   8   0  -6   1  -6  -3  -5
   0   0  -3   0   6   2   2   0   0  -2  -3   1   1   2   4   5
  -7  -9  -3  -4  -4  -5   1   8  -3  -6   3   5  -2   4  -1   0
The a(2)=9 solutions (w,x,y) of w+2x+3y=0, as described in the Comments section, are as follows: (-2,-2,2), (-2,1,0), (-1,-1,1), (-1,2,-1), (0,0,0), (1,-2,1), (1,1,-1), (2,-1,0), (2,2,-2). - _Clark Kimberling_, Apr 10 2012
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210

Crossrefs

Cf. A202252.

Programs

Mathematica

t[n_]:=t[n]=Flatten[Table[w+2x+3y,{w,-n,n},
{x,-n,n},{y,-n,n}]]
c[n_]:=Count[t[n],0]
t=Table[c[n],{n,1,50}] (* A143978 ? *)
(t-1)/2 (* A143978 *)
(* Clark Kimberling, Apr 10 2012 *)

Formula

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

Conjecture: a(n) = 1+2*floor((2*n^2+2*n)/3). - Clark Kimberling, Apr 12 2012

Empirical g.f.: x*(3 + 3*x + 2*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Mar 03 2018