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A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.

%I A212069 #28 Aug 09 2025 07:40:49
%S A212069 0,1,2,9,22,41,72,115,170,243,334,443,576,733,914,1125,1366,1637,1944,
%T A212069 2287,2666,3087,3550,4055,4608,5209,5858,6561,7318,8129,9000,9931,
%U A212069 10922,11979,13102,14291,15552,16885,18290,19773,21334,22973
%N A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.
%C A212069 w is the average of {x,y,z}, as well as {w,x,y,z}.
%C A212069 For a guide to related sequences, see A211795.
%C A212069 a(n) is also the number of (w,x,y,z) with all terms in {0,1,...,n-1} and 3*w = x+y+z. - _Clark Kimberling_, May 16 2012
%H A212069 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A212069 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 A212069 From _R. J. Mathar_, Jun 25 2012: (Start)
%F A212069 G.f.: x*(1 - x + 6*x^2 - x^3 + x^4)/((1 + x + x^2)*(1 - x)^4).
%F A212069 a(n) = (n^3 + 2*A049347(n-1))/3. (End)
%F A212069 E.g.f.: (3*exp(x)*x*(1 + 3*x + x^2) + 4*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - _Stefano Spezia_, Aug 06 2025
%t A212069 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212069 (Do[If[3 w == x + y + z, s = s + 1],
%t A212069 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212069 Map[t[#] &, Range[0, 50]] (* A212087 *)
%t A212069 FindLinearRecurrence[%]
%t A212069 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A212069 LinearRecurrence[{3, -3, 2, -3, 3, -1},{0, 1, 2, 9, 22, 41},42] (* _Ray Chandler_, Aug 02 2015 *)
%Y A212069 Cf. A049347, A211795.
%K A212069 nonn,easy
%O A212069 0,3
%A A212069 _Clark Kimberling_, May 01 2012