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A212251 Number of (w,x,y,z) with all terms in {1,...,n} and 3w = x + y + z + n + 1.

%I A212251 #16 Dec 02 2017 08:47:54
%S A212251 0,0,1,6,15,30,54,87,131,189,261,349,456,582,729,900,1095,1316,1566,
%T A212251 1845,2155,2499,2877,3291,3744,4236,4769,5346,5967,6634,7350,8115,
%U A212251 8931,9801,10725,11705,12744,13842,15001,16224,17511,18864,20286,21777,23339,24975
%N A212251 Number of (w,x,y,z) with all terms in {1,...,n} and 3w = x + y + z + n + 1.
%C A212251 Also, the number of (w,x,y,z) with all terms in {1,...,n} and 3w = x + y + z - n - 1.
%C A212251 For a guide to related sequences, see A211795.
%H A212251 Colin Barker, <a href="/A212251/b212251.txt">Table of n, a(n) for n = 0..1000</a>
%H A212251 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A212251 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
%F A212251 G.f.: x^2*(1 + 3*x + x^3) / ((1 - x)^4*(1 + x + x^2)). - _Colin Barker_, Dec 02 2017
%t A212251 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212251 (Do[If[3 w == x + y + z + n + 1, s = s + 1],
%t A212251 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212251 Map[t[#] &, Range[0, 40]]  (* A212251 *)
%t A212251 (* _Peter J. C. Moses_, Apr 13 2012 *)
%o A212251 (PARI) concat(vector(2), Vec(x^2*(1 + 3*x + x^3) / ((1 - x)^4*(1 + x + x^2)) + O(x^40))) \\ _Colin Barker_, Dec 02 2017
%Y A212251 Cf. A211795.
%K A212251 nonn,easy
%O A212251 0,4
%A A212251 _Clark Kimberling_, May 15 2012