

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
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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, im+1)..i, max(1, jm+1)..min(#g, j+m1)])), 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



