A212892 - OEIS

%I #51 Sep 12 2023 13:38:44

%S 0,0,2,8,32,72,162,288,512,800,1250,1800,2592,3528,4802,6272,8192,

%T 10368,13122,16200,20000,24200,29282,34848,41472,48672,57122,66248,

%U 76832,88200,101250,115200,131072,147968,167042,187272,209952,233928,260642,288800,320000,352800,388962,426888

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

%C 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.

%C 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 Davies's work went unpublished until it appeared in the survey paper by Drakakis.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

%C Every term is even.

%H N. J. A. Sloane, <a href="/A212892/b212892.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Drakakis, <a href="http://projecteuclid.org/euclid.jam/1168975519">A review of Costas arrays</a>, Journal of Applied Mathematics, pp. 1-32, 2006, Article ID 26385.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).

%F 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.

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

%F a(n) = 2*floor(n^2/4)^2 = 2*A030179(n). - _Ridouane Oudra_, Sep 12 2023

%p 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

%t (* If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... *)

%t 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)]];

%t m = Map[t[#] &, Range[0, 40]]   (* A212892 *)

%t m/2 (* integers *)

%Y Cf. A211795.

%Y Cf. A030179.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 30 2012, and _Bill Correll, Jr._, Jun 07 2014

%E 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