cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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
Offset: 1

Views

Author

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

Keywords

Comments

Equivalently, no pair of chosen cells are closer than a knight's move apart. This is a one-error-correcting code in the Lee metric.
Equivalently, maximal number of 5-celled 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) = ceiling(n^2/5). - Rob Pratt, Jul 10 2015
Conjecture holds for n <= 70. - Giovanni Resta, Jul 29 2015

Examples

			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.

Links

Crossrefs

Main diagonal of A085576.
Cf. A233735.

Programs

  • 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[n-4] - a[n-3] - a[n-2] - a[n-1]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 24 2016 *)

Formula

a(n) approaches n^2/5 as n -> infinity.
From Colin Barker, Oct 15 2016: (Start)
Conjectures:
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n > 8.
G.f.: x*(1 - x + x^2 + x^3 - x^5 + x^6 - x^9 + 2*x^10 - x^11) / ((1-x)^3*(1 + x + x^2 + x^3 + x^4)). (End)

Extensions

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

A233735 G.f.: x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) * (1-x)^2).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 8, 10, 13, 16, 20, 25, 29, 34, 39, 45, 52, 58, 65, 72, 80, 88, 96, 105, 114, 124, 134, 144, 155, 166, 178, 190, 202, 215, 228, 242, 256, 270, 285, 300, 316, 332, 348, 365, 382, 400, 418, 436, 455, 474, 494, 514, 534
Offset: 0

Views

Author

Kival Ngaokrajang, Dec 15 2013

Keywords

Comments

The second differences repeat with period 1,0,1,0,0 for n >= 20.
a(n) is a lower bound on A085577(n-2). The Ngaokrajang link shows arrangements of a(n) Greek crosses in an n X n grid. Note that a(11)=16, whereas A085577(9)=17, so the bound is not always tight. - N. J. A. Sloane, Apr 19 2015

Links

Crossrefs

Cf. A085576, A085577, A233035, A233036.

Programs

  • Mathematica
    CoefficientList[Series[x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 08 2018 *)
  • PARI
    x='x+O('x^50); Vec(x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2)) \\ G. C. Greubel, Jan 08 2018

Extensions

Entry revised by N. J. A. Sloane, Apr 19 2015. The new definition is a g.f. found by Ralf Stephan on Dec 17 2013. The old definition was wrong.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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