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%I #17 Jan 09 2024 16:36:26
%S 1,4,17,20,53,56,117,120,217,220,361,364,557,560,813,816,1137,1140,
%T 1537,1540,2021,2024,2597,2600,3273,3276,4057,4060,4957,4960,5981,
%U 5984,7137,7140,8433,8436,9877,9880,11477,11480,13241,13244,15177
%N Number of (w,x,y,z) with all terms in {0,...,n} and w=2*floor((x+y+z)/2).
%C For a guide to related sequences, see A211795.
%H Colin Barker, <a href="/A212748/b212748.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
%F G.f.: (1+3*x+10*x^2-6*x^3-3*x^4+3*x^5) / ((1+x)^3*(x-1)^4).
%F From _Colin Barker_, Jan 29 2016: (Start)
%F a(n) = (2*n^3+3*((-1)^n+5)*n^2+(15*(-1)^n+37)*n+12)/12.
%F a(n) = (n^3+9*n^2+26*n+6)/6 for n even.
%F a(n) = (n^3+6*n^2+11*n+6)/6 for n odd. (End)
%t t = Compile[{{n, _Integer}}, Module[{s = 0},
%t (Do[If[w == 2*Floor[(x + y + z)/2], s = s + 1],
%t {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t Map[t[#] &, Range[0, 45]] (* A212748 *)
%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,17,20,53,56,117},50] (* _Harvey P. Dale_, Jun 08 2018 *)
%o (PARI) Vec((1+3*x+10*x^2-6*x^3-3*x^4+3*x^5)/((1+x)^3*(x-1)^4) + O(x^100)) \\ _Colin Barker_, Jan 29 2016
%Y Cf. A211795.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, May 28 2012
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