A212563 - OEIS

%I #10 Aug 28 2020 11:10:29

%S 0,1,16,78,240,577,1182,2172,3680,5865,8900,12986,18336,25193,33810,

%T 44472,57472,73137,91800,113830,139600,169521,204006,243508,288480,

%U 339417,396812,461202,533120,613145,701850,799856,907776,1026273,1156000,1297662,1451952

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

%C For a guide to related sequences, see A211795.

%H Colin Barker, <a href="/A212563/b212563.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

%F 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 From _Colin Barker_, Dec 10 2015: (Start)

%F a(n) = 1/96*(82*n^4+36*n^3-16*n^2-6*((-1)^n-1)*n+9*((-1)^n-1)).

%F G.f.: x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2).

%F (End)

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[w + x <= 2 y + 2 z, s = s + 1],

%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

%t Map[t[#] &, Range[0, 40]]   (* A212563 *)

%t LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,16,78,240,577,1182},40] (* _Harvey P. Dale_, Aug 28 2020 *)

%o (PARI) concat(0, Vec(x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2) + O(x^50))) \\ _Colin Barker_, Dec 10 2015

%Y Cf. A211795.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 21 2012