A212145 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x+y+z.

0, 1, 16, 81, 255, 621, 1285, 2377, 4050, 6481, 9870, 14441, 20441, 28141, 37835, 49841, 64500, 82177, 103260, 128161, 157315, 191181, 230241, 275001, 325990, 383761, 448890, 521977, 603645, 694541, 795335, 906721, 1029416, 1164161

Offset: 0

Author

Clark Kimberling, May 02 2012

Keywords

• nonn
• easy

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4, -5, 0, 5, -4, 1).

Crossrefs

Cf. A211795.

Programs

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

Maple

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

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]] (* A212145 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
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 *)
LinearRecurrence[{4, -5, 0, 5, -4, 1},{0, 1, 16, 81, 255, 621},34] (* Ray Chandler, Aug 02 2015 *)

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

Formula

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

a(n) = (3-3*(-1)^n-8*n-4*n^2+8*n^3+94*n^4)/96. - Colin Barker, Nov 21 2014

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