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.

Showing 1-3 of 3 results.

A085576 Array read by antidiagonals: T(n,k) = size of maximal subset of nodes in n X k grid such that there at least 3 edges between any pair of nodes (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 4, 4, 3, 3, 4, 5, 6, 6, 6, 5, 4, 3, 4, 5, 6, 6, 7, 7, 6, 6, 5, 4, 4, 5, 6, 8, 8, 8, 8, 8, 6, 5, 4, 4, 6, 7, 8, 9, 9, 9, 9, 8, 7, 6, 4
Offset: 1

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Author

N. J. A. Sloane, Jul 08 2003

Keywords

Comments

The 1-neighborhoods of the nodes must be disjoint: i.e. this is a 1-error correcting code.

Examples

			Array begins
1 1 1 2 2 2 3 3 3 4 ...
1 1 2 2 3 3 4 4 5 5 ...
1 2 2 3 4 4 5 6 6 7 ...
2 2 3 4 5 6 6 8 8 9 ...
For example, T(3,4) = 3 (*'s indicate the chosen nodes):
o--*--o--o
|..|..|..|
o--o--o--o
|..|..|..|
*--o--o--*

Crossrefs

Main diagonal gives A085577.

Formula

T(n, 1) = floor((n+2)/3), T(n, 2) = floor((n+1)/2).

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.

A354673 Smallest number of unit cells that must be removed from an n X n square board in order to avoid any cycles.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 13, 18, 22, 28, 34, 42, 49, 58, 66, 76, 86, 98, 109, 122, 134, 148, 162, 178, 193, 210, 226, 244, 262, 282, 301, 322, 342, 364, 386, 410, 433, 458, 482, 508, 534, 562, 589, 618, 646, 676, 706, 738, 769, 802, 834, 868, 902, 938, 973, 1010, 1046
Offset: 1

Views

Author

Giedrius Alkauskas, Jun 02 2022

Keywords

Comments

A "cycle" means a rook-connected closed path of squares.
The proof of this result is given in the Links section.
a(n+1) is very close to A239231(n); more precisely, the difference is the sequence 1,0,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,3,2.

Examples

			For n = 2, a(2) = 1, since removing any unit square from the 2 X 2 board leaves no cycles.
For n = 5, a(5) = 6 removed unit squares can be arranged as follows:
  x****
  *x*x*
  **x**
  *x*x*
  *****

Links

Crossrefs

Cf. A085577, A239072, A085577, A104519, A000170, A239231.

Formula

a(n) = ceiling(n^2/3 - n/6 + 4/3) - ceiling(n/2) for n >= 3.
From Stefano Spezia, Jun 02 2022: (Start)
G.f.: x^2*(1 + x^2 + 2*x^4 - x^5 + x^6 - x^7 + x^8)/((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n > 2. (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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