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A292673 Least number of symbols required to fill a grid of size n X n row by row in the greedy way such that in any row or column or rectangular 3 X 3 block no symbol occurs twice. 5
1, 4, 9, 11, 13, 13, 13, 13, 14, 14, 15, 17, 18, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 38, 39, 40, 42, 45, 47, 49, 51, 53, 54, 55, 55, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 74, 76, 78, 79, 83, 83, 85, 86, 88, 90, 91, 92, 93, 96 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Consider the symbols as positive integers. By the greedy way we mean to fill the grid row by row from left to right always with the least possible positive integer such that the three constraints (on rows, columns and rectangular blocks) are satisfied.

In contrast to the sudoku case, the 3 X 3 rectangles have "floating" borders, so the constraint is actually equivalent to say that any element must be different from all neighbors in a Moore neighborhood of range 2 (having up to 5*5 = 25 grid points).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Moore Neighborhood

EXAMPLE

For n = 4, the 4 X 4 grid is filled as follows (using hexadecimal digits):

   [1  2  3  4]

   [4  5  6  1]

   [7  8  9  A]

   [2  3  B  7], whence a(4) = # { 1, ..., 9, A, B} = 11.

For n = 8, the grid is filled as follows:

  [1  2  3  4  5  6  7  8]

  [4  5  6  1  2  3  9  A]

  [7  8  9  A  B  C  1  2]

  [2  3  C  7  4  5  6  B]

  [5  1  D  2  3  8  A  4]

  [6  4  8  9  1  D  2  3]

  [3  7  A  5  6  B  C  1]

  [9  B  2  3  7  4  5  6], whence a(8) = # { 1, ..., 9, A, B, C, D } = 13.

For n = 5, 6 and 7, the solution is just the upper left n X n part of the above grid: all of these also require 13 symbols.

PROG

(PARI) a(n, m=3, g=matrix(n, n))={my(ok(g, k, i, j, m)=if(m, ok(g[i, ], k)&&ok(g[, j], k)&&ok(concat(Vec(g[max(1, i-m+1)..i, max(1, j-m+1)..min(#g, j+m-1)])), k), !setsearch(Set(g), k))); for(i=1, n, for(j=1, n, for(k=1, n^2, ok(g, k, i, j, m)&&(g[i, j]=k)&&break))); vecmax(g)} \\ without "vecmax" the program returns the full n X n board.

CROSSREFS

Cf. A292670, A292671, A292672, ..., A292679.

Sequence in context: A291626 A312833 A243651 * A035233 A341788 A010396

Adjacent sequences:  A292670 A292671 A292672 * A292674 A292675 A292676

KEYWORD

nonn

AUTHOR

M. F. Hasler, Sep 20 2017

EXTENSIONS

Terms a(40) and beyond from Andrew Howroyd, Feb 22 2020

STATUS

approved

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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)