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A212562 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<2y+2z.

%I A212562 #11 Sep 08 2022 08:46:02
%S A212562 0,1,15,73,228,551,1137,2097,3568,5701,8675,12681,17940,24683,33173,
%T A212562 43681,56512,71977,90423,112201,137700,167311,201465,240593,285168,
%U A212562 335661,392587,456457,527828,607251,695325,792641,899840,1017553,1146463,1287241,1440612
%N A212562 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<2y+2z.
%C A212562 For a guide to related sequences, see A211795.
%H A212562 Colin Barker, <a href="/A212562/b212562.txt">Table of n, a(n) for n = 0..1000</a>
%H A212562 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212562 a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).
%F A212562 From _Colin Barker_, Dec 05 2015: (Start)
%F A212562 a(n) = 1/96*(82*n^4 + 12*n^3 + 8*n^2 + 6*((-1)^n-1)*n - 3*(-1)^n + 3).
%F A212562 G.f.: x*(1 + 12*x + 29*x^2 + 29*x^3 + 10*x^4 + x^5) / ((1-x)^5*(1+x)^2). (End)
%t A212562 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212562 (Do[If[w + x < 2 y + 2 z, s = s + 1],
%t A212562 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212562 Map[t[#] &, Range[0, 40]]   (* A212562 *)
%t A212562 CoefficientList[Series[x (1 + 12 x + 29 x^2 + 29 x^3 + 10 x^4 + x^5)/((1 - x)^5 (1 + x)^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Dec 05 2015 *)
%o A212562 (PARI) concat(0, Vec(x*(1+12*x+29*x^2+29*x^3+10*x^4+x^5)/((1-x)^5*(1+x)^2) + O(x^100))) \\ _Colin Barker_, Dec 05 2015
%o A212562 (Magma) I:=[0,1,15,73,228,551,1137]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)-5*Self(n-3)+5*Self(n-4)+Self(n-5)-3*Self(n-6)+Self(n-7): n in [1..40]]; // _Vincenzo Librandi_, Dec 05 2015
%Y A212562 Cf. A211795.
%K A212562 nonn,easy
%O A212562 0,3
%A A212562 _Clark Kimberling_, May 21 2012