A211626 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+4x+4y>0.

0, 4, 32, 108, 250, 492, 854, 1360, 2021, 2885, 3965, 5285, 6849, 8719, 10901, 13419, 16270, 19530, 23198, 27298, 31820, 36854, 42392, 48458, 55035, 62227, 70019, 78435, 87451, 97185, 107615, 118765, 130604, 143264, 156716, 170984, 186030, 202000, 218858

Offset: 0

Author

Clark Kimberling, Apr 17 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + 4 x + 4 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211626 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1},{0,4,32,108,250,492,854,1360,2021,2885},40] (* Harvey P. Dale, Nov 29 2013 *)

PARI

concat(0, Vec(x*(4 + 24*x + 48*x^2 + 66*x^3 + 92*x^4 + 72*x^5 + 48*x^6 + 23*x^7 + 7*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) - 4*a(n-5) + 2*a(n-6) - a(n-8) + 2*a(n-9) - a(n-10) for n>9.

G.f.: x*(4 + 24*x + 48*x^2 + 66*x^3 + 92*x^4 + 72*x^5 + 48*x^6 + 23*x^7 + 7*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)^2). - Colin Barker, Dec 05 2017