A212892 a(n) = n^4/8 if n is even, a(n) = (n^2-1)^2/8 if n is odd.

0, 0, 2, 8, 32, 72, 162, 288, 512, 800, 1250, 1800, 2592, 3528, 4802, 6272, 8192, 10368, 13122, 16200, 20000, 24200, 29282, 34848, 41472, 48672, 57122, 66248, 76832, 88200, 101250, 115200, 131072, 147968, 167042, 187272, 209952, 233928, 260642, 288800, 320000, 352800, 388962, 426888

Offset: 0

Author

Clark Kimberling, May 30 2012, and Bill Correll, Jr., Jun 07 2014

Keywords

• nonn
• easy

Comments

The sequence a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... arises as the number of quadruples (w,x,y,z) with all terms in {0,...,n} and w-x, x-y, and y-z all odd. For a guide to related sequences, see A211795.

The sequence a(3)=2, a(4)=8, a(5)=32, ... is the number of L3 configurations in an n X n permutation array. An L3 configuration is defined to be a set of 3 equally-spaced, collinear points in a permutation array. L3 configurations were first enumerated by Davies in his study of the density of Costas arrays. They constitute a violation of the definition of a Costas array, so Costas arrays cannot have any. Davies's work went unpublished until it appeared in the survey paper by Drakakis. - Bill Correll, Jr., Jun 07 2014

Every term is even.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• K. Drakakis, A review of Costas arrays, Journal of Applied Mathematics, pp. 1-32, 2006, Article ID 26385.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A211795.

Cf. A030179.

Programs

Maple

f:=n->if n mod 2 = 0 then n^4/8 else (n^2-1)^2/8; fi; [seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 06 2015

Mathematica

(* If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... *)
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Mod[w - x, 2] == Mod[x - y, 2] == Mod[y - z, 2] == 1, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212892 *)
m/2 (* integers *)

Formula

If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ...: a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8). G.f.: f(x)/g(x), where f(x) = x+2*x^2+6*x^3+2*x^4+x^5 and g(x) = ((1-x)^5)*(1+x)^3.

Sum_{n>=2} 1/a(n) = Pi^4/180 + Pi^2/6 - 3/2. - Amiram Eldar, Sep 08 2022

a(n) = 2*floor(n^2/4)^2 = 2*A030179(n). - Ridouane Oudra, Sep 12 2023

Extensions

Formed by merging two entries that arose in different contexts. Thanks to Alois P. Heinz, Mar 04 2015 for noticing that the sequences were essentially identical. - N. J. A. Sloane, Mar 06 2015