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A292670 Least number of symbols required to fill a grid of size n^2 X n^2 row by row in the greedy way such that in no row or column or n X n square any symbol occurs twice. 9
1, 6, 14, 26, 40, 53, 73, 114 (list; graph; refs; listen; history; text; internal format)



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 n X n 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 n-1 (having up to (2n-1)^2 grid points).


Table of n, a(n) for n=1..8.

Eric W. Weisstein, Moore Neighborhood, on MathWorld--A Wolfram Web Resource.


For n = 2, the 4 X 4 grid is filled as follows:

   [1 2 3 4]

   [3 4 1 2]

   [2 5 6 3]

   [4 1 2 5], whence a(2) = 6.


(PARI) a(m, n=m^2, 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^2 X n^2 board.


Cf. A292671: grid size independent of block size; A292672, ..., A292679: A(m,n) for fixed n=2,...,9).

Sequence in context: A026055 A288939 A165986 * A131951 A168648 A093776

Adjacent sequences:  A292667 A292668 A292669 * A292671 A292672 A292673




M. F. Hasler, Sep 20 2017



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Last modified September 19 10:53 EDT 2021. Contains 347556 sequences. (Running on oeis4.)