

A085577


Size of maximal subset of the n^2 cells in an n X n grid such that there are at least 3 edges between any pair of chosen cells.


3



1, 1, 2, 4, 6, 8, 10, 13, 17, 20, 25, 29, 34, 40, 45, 52, 58, 65, 73, 80, 89, 97, 106, 116, 125, 136, 146, 157, 169, 180, 193, 205, 218, 232, 245, 260, 274, 289, 305, 320, 337, 353, 370, 388, 405, 424, 442, 461, 481, 500, 521, 541, 562, 584, 605, 628, 650
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OFFSET

1,3


COMMENTS

Equivalently, no pair of chosen cells are closer than a knight's move apart. This is a oneerrorcorrecting code in the Lee metric.
Equivalently, maximal number of 5celled Greek crosses that can be packed into an n+2 X n+2 chessboard.
A233735(n+2) is a lower bound on a(n).
Conjecture: if n == 4 (mod 5), then a(n)=(n^2+4)/5.  Erich Friedman, Apr 19 2015
More general conjecture: if n != 5, then a(n) = ceil(n^2/5).  Rob Pratt, Jul 10 2015
Conjecture holds for n <= 70.  Giovanni Resta, Jul 29 2015


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..70
Kival Ngaokrajang, Packings of A233735(n) Greek crosses. [Note that it is possible to pack 17 Greek crosses into an 11 X 11 grid (see EXAMPLES), so these arrangements are not always optimal.]
Popular Computing (Calabasas, CA), Problem 175: Knights Away, Vol. 5, (No. 50, May 1977), pp. PC5018 to PC5019.


FORMULA

a(n) approaches n^2/5 as n > infinity.
From Colin Barker, Oct 15 2016: (Start)
Conjectures:
a(n) = 2*a(n1)a(n2)+a(n5)2*a(n6)+a(n7) for n>8.
G.f.: x*(1x+x^2+x^3x^5+x^6x^9+2*x^10x^11) / ((1x)^3*(1+x+x^2+x^3+x^4)). (End)


EXAMPLE

For example, a(3) = 2:
..o
...
o..
a(9)=17 (from Erich Friedman, Apr 18 2015):
.o....o..
...o....o
o....o...
..o....o.
....o....
.o....o..
...o....o
o....o...
..o....o.


MATHEMATICA

(* Warning: this program gives correct results up to n=70, but must not be used to extend the sequence beyond that limit. *) a[n_] := a[n] = If[n <= 9, {1, 1, 2, 4, 6, 8, 10, 13, 17}[[n]], n^2  4*n + 8  a[n4]  a[n3]  a[n2]  a[n1]]; Table[a[n], {n, 1, 70}] (* JeanFrançois Alcover, Nov 24 2016 *)


CROSSREFS

Main diagonal of A085576.
Cf. A233735.
Sequence in context: A056827 A024172 A233735 * A121832 A253241 A302979
Adjacent sequences: A085574 A085575 A085576 * A085578 A085579 A085580


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Jul 08 2003; entry revised Apr 19 2015


EXTENSIONS

a(14)a(30) from Rob Pratt, Jul 10 2015
a(31)a(57) from Giovanni Resta, Jul 29 2015


STATUS

approved



