cp's OEIS Frontend

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

%I A202253 #19 Mar 03 2018 05:35:55
%S A202253 3,9,17,27,41,57,75,97,121,147,177,209,243,281,321,363,409,457,507,
%T A202253 561,617,675,737,801,867,937,1009,1083,1161,1241,1323,1409,1497,1587,
%U A202253 1681,1777,1875,1977,2081,2187,2297,2409,2523,2641,2761,2883,3009,3137,3267
%N A202253 Number of zero-sum -n..n arrays of 3 elements with adjacent element differences also in -n..n.
%C A202253 Row 3 of A202252.
%C A202253 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
%H A202253 R. H. Hardin, <a href="/A202253/b202253.txt">Table of n, a(n) for n = 1..210</a>
%F A202253 Empirical: a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
%F A202253 Conjecture: a(n) = 1+2*floor((2*n^2+2*n)/3). - _Clark Kimberling_, Apr 12 2012
%F A202253 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
%e A202253 Some solutions for n=10:
%e A202253    7   9   6   4  -2   3  -3  -8   3   8   0  -6   1  -6  -3  -5
%e A202253    0   0  -3   0   6   2   2   0   0  -2  -3   1   1   2   4   5
%e A202253   -7  -9  -3  -4  -4  -5   1   8  -3  -6   3   5  -2   4  -1   0
%e A202253 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
%t A202253 t[n_]:=t[n]=Flatten[Table[w+2x+3y,{w,-n,n},
%t A202253 {x,-n,n},{y,-n,n}]]
%t A202253 c[n_]:=Count[t[n],0]
%t A202253 t=Table[c[n],{n,1,50}] (* A143978 ? *)
%t A202253 (t-1)/2 (* A143978 *)
%t A202253 (* _Clark Kimberling_, Apr 10 2012 *)
%Y A202253 Cf. A202252.
%K A202253 nonn
%O A202253 1,1
%A A202253 _R. H. Hardin_, Dec 14 2011