A292678 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 8 X 8 block no symbol occurs twice.

1, 4, 9, 16, 25, 36, 49, 64, 66, 66, 68, 68, 69, 69, 70, 70, 73, 76, 81, 84, 83, 82, 84, 79, 81, 83, 85, 85, 85, 85, 86, 88, 88, 92, 91, 88, 89, 92, 89, 93, 92, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 114, 114, 114, 114, 114, 114

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 8 X 8 rectangles have "floating" borders, so the constraint is actually equivalent to say that an element must be different from all neighbors in a Moore neighborhood of range 7 (having up to 15*15 = 225 grid points).

See sequences A292672, A292673, A292674 for examples.

Links

Table of n, a(n) for n=1..70.

Eric Weisstein's World of Mathematics, Moore Neighborhood

Prog

(PARI) a(n, m=8, 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.
(Python) # uses function in A292673
print([A292673(n, b=8) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

Crossrefs

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

Sequence in context: A028907 A361265 A292677 * A072595 A334832 A169669

Adjacent sequences: A292675 A292676 A292677 * A292679 A292680 A292681

Keyword

nonn

Author

M. F. Hasler, Sep 20 2017

Extensions

Terms a(60) and beyond from Andrew Howroyd, Feb 22 2020

Status

approved