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%I #44 Jun 10 2023 08:09:08
%S 0,3,8,16,27,40,56,75,96,120,147,176,208,243,280,320,363,408,456,507,
%T 560,616,675,736,800,867,936,1008,1083,1160,1240,1323,1408,1496,1587,
%U 1680,1776,1875,1976,2080,2187,2296,2408,2523,2640,2760,2883
%N Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w + 2x + 3y = 1.
%C For a guide to related sequences, see A211422.
%C For n>2, this is the number of 1's in the partitions of 4n-4 into 4 parts. - _Wesley Ivan Hurt_, Mar 13 2014
%C List of triples: [4*k*(3*k-1), 4*k*(3*k+1), 3*(2*k+1)^2], respectively A014642, 8*A005449, 3*A016754. - _Luce ETIENNE_, May 31 2017
%C Conjecture: Number of partitions of 4n+2 into 3 parts. - _George Beck_, Mar 23 2023
%H Robert Israel, <a href="/A211480/b211480.txt">Table of n, a(n) for n = 0..1000</a>
%F Conjectures from _Colin Barker_, May 15 2017: (Start)
%F G.f.: x^2*(3 + 2*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)).
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>=5.
%F (End)
%F Conjecture: a(n) = (8*floor(n/3)*(2*n-3*floor(n/3)-1)+3*(1-(-1)^(n+2-floor((n+2)/3))))/2 = floor((2*n-1)^2/3). - _Luce ETIENNE_, May 25 2017
%p f:= proc(n) local x,r,ymin,ymax;
%p r:= 0:
%p for x from -n to n do
%p ymin:= max(-n, ceil((-n+1-2*x)/3));
%p ymax:= min(n, floor((n+1-2*x)/3));
%p if ymin <= ymax then r:= r + ymax-ymin+1 fi
%p od;
%p r
%p end proc:
%p map(f, [$0..50]); # _Robert Israel_, Jun 09 2023
%t t[n_] := t[n] = Flatten[Table[w + 2 x + 3 y - 1, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211480 *)
%t b[0] := 0; b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2])/(4 n), {i, 0, 2 n}]; Table[b[n - 1] + 2 (n - 1), {n, 50}] (* _Wesley Ivan Hurt_, Mar 13 2014 *)
%Y Cf. A211422.
%Y Cf. A005449, A014642, A016754.
%K nonn
%O 0,2
%A _Clark Kimberling_, Apr 12 2012
%E Offset corrected by _Robert Israel_, Jun 09 2023
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