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A213397 Number of (w,x,y) with all terms in {0,...,n} and 2*w >= |x+y-z|.

%I A213397 #20 Dec 23 2017 04:48:49
%S A213397 1,5,18,43,83,144,229,341,486,667,887,1152,1465,1829,2250,2731,3275,
%T A213397 3888,4573,5333,6174,7099,8111,9216,10417,11717,13122,14635,16259,
%U A213397 18000,19861,21845,23958,26203,28583,31104,33769,36581,39546,42667,45947,49392,53005,56789,60750
%N A213397 Number of (w,x,y) with all terms in {0,...,n} and 2*w >= |x+y-z|.
%C A213397 For a guide to related sequences, see A212959.
%H A213397 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A213397 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 A213397 G.f.: (1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x + x^2)).
%F A213397 a(n) = (n+1)^3 - A213396(n).
%F A213397 a(n) = floor(2*n^3/3) + 2*n*(n + 1) + 1. - _Bruno Berselli_, Dec 22 2017
%t A213397 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A213397 (Do[If[2 w >= Abs[x + y - w], s = s + 1],
%t A213397 {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A213397 m = Map[t[#] &, Range[0, 60]]   (* A212297 *)
%t A213397 CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 - x)^4*(1 + x + x^2)), {x, 0, 44}], x] (* _Michael De Vlieger_, Dec 22 2017 *)
%o A213397 (PARI) first(n) = Vec((1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x + x^2)) + O(x^n)) \\ _Iain Fox_, Dec 22 2017
%Y A213397 Cf. A212959, A213396.
%K A213397 nonn,easy
%O A213397 0,2
%A A213397 _Clark Kimberling_, Jun 12 2012