A212145 - OEIS

%I #18 Sep 08 2022 08:46:02

%S 0,1,16,81,255,621,1285,2377,4050,6481,9870,14441,20441,28141,37835,

%T 49841,64500,82177,103260,128161,157315,191181,230241,275001,325990,

%U 383761,448890,521977,603645,694541,795335,906721,1029416,1164161

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

%C A212145(n)+A212087(n)=4^n. For a guide to related sequences, see A211795.

%H Colin Barker, <a href="/A212145/b212145.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4, -5, 0, 5, -4, 1).

%F a(n) = 4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6).

%F a(n) = (3-3*(-1)^n-8*n-4*n^2+8*n^3+94*n^4)/96. - _Colin Barker_, Nov 21 2014

%F G.f.: -x*(x^4+11*x^3+22*x^2+12*x+1) / ((x-1)^5*(x+1)). - _Colin Barker_, Nov 21 2014

%p A212145:=n->(3-3*(-1)^n-8*n-4*n^2+8*n^3+94*n^4)/96: seq(A212145(n), n=0..40); # _Wesley Ivan Hurt_, Nov 21 2014

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

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

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

%t Map[t[#] &, Range[0, 60]] (* A212145 *)

%t FindLinearRecurrence[%]

%t (* _Peter J. C. Moses_, Apr 13 2012 *)

%t CoefficientList[Series[x (x^4 + 11 x^3 + 22 x^2 + 12 x + 1) / ((1 - x)^5 (x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 22 2014 *)

%t LinearRecurrence[{4, -5, 0, 5, -4, 1},{0, 1, 16, 81, 255, 621},34] (* _Ray Chandler_, Aug 02 2015 *)

%o (PARI) concat(0, Vec(-x*(x^4+11*x^3+22*x^2+12*x+1)/((x-1)^5*(x+1)) + O(x^100))) \\ _Colin Barker_, Nov 21 2014

%o (Magma) [(3-3*(-1)^n-8*n-4*n^2+8*n^3+94*n^4)/96 : n in [0..40]]; // _Wesley Ivan Hurt_, Nov 21 2014

%Y Cf. A211795.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 02 2012