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A212247 Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z+n.

%I A212247 #16 Jul 06 2021 09:57:11
%S A212247 0,1,4,13,29,56,95,150,222,315,430,571,739,938,1169,1436,1740,2085,
%T A212247 2472,2905,3385,3916,4499,5138,5834,6591,7410,8295,9247,10270,11365,
%U A212247 12536,13784,15113,16524,18021,19605,21280,23047,24910,26870,28931
%N A212247 Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z+n.
%C A212247 For a guide to related sequences, see A211795.
%H A212247 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, -2, 3, -1).
%F A212247 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 A212247 G.f.: x*(1+x+3*x^2)/((1+x)*(1-x)^4). [_Bruno Berselli_, May 30 2012]
%F A212247 a(n) = (2*n*(10*n^2+3*n+2)-9(-1)^n+9)/48. [_Bruno Berselli_, May 30 2012]
%t A212247 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212247 (Do[If[2 w == x + y + z - n, s = s + 1],
%t A212247 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212247 Map[t[#] &, Range[0, 60]]  (* A212246 *)
%t A212247 FindLinearRecurrence[%]
%t A212247 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A212247 LinearRecurrence[{3, -2, -2, 3, -1},{0, 1, 4, 13, 29},42] (* _Ray Chandler_, Aug 02 2015 *)
%t A212247 CoefficientList[Series[x (1+x+3x^2)/((1+x)(1-x)^4),{x,0,50}],x] (* _Harvey P. Dale_, Jul 06 2021 *)
%Y A212247 Cf. A211795.
%K A212247 nonn,easy
%O A212247 0,3
%A A212247 _Clark Kimberling_, May 09 2012