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A202253 Number of zero-sum -n..n arrays of 3 elements with adjacent element differences also in -n..n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
LINKS
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
EXAMPLE
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
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 *)
CROSSREFS
Cf. A202252.
Sequence in context: A329422 A004621 A014209 * A057258 A018466 A171152
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 14 2011
STATUS
approved

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Last modified February 20 21:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)