A212571 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|<|x-y|+|y-z|.

0, 0, 8, 48, 168, 428, 916, 1728, 2992, 4840, 7440, 10960, 15608, 21588, 29148, 38528, 50016, 63888, 80472, 100080, 123080, 149820, 180708, 216128, 256528, 302328, 354016, 412048, 476952, 549220, 629420, 718080, 815808, 923168, 1040808, 1169328, 1309416

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

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 (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A211795, A212570.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] < Abs[x - y] + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212571 *)
%/4  (* integers *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,8,48,168,428,916},40] (* Harvey P. Dale, Aug 28 2024 *)

PARI

concat([0,0], Vec(4*x^2*(2+6*x+8*x^2+3*x^3)/((1-x)^5*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 05 2015

Formula

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).

From Colin Barker, Dec 05 2015: (Start)

a(n) = 1/48*(38*n^4-20*n^3-32*n^2+2*(3*(-1)^n+13)*n+3*((-1)^n-1)).

G.f.: 4*x^2*(2+6*x+8*x^2+3*x^3) / ((1-x)^5*(1+x)^2). (End)

E.g.f.: (x*(3 + 87*x + 104*x^2 + 19*x^3)*cosh(x) - (3 - 9*x - 87*x^2 - 104*x^3 - 19*x^4)*sinh(x))/24. - Stefano Spezia, Mar 12 2025