A292671 Upper right triangle A(m,n) = least number of symbols required to fill a grid of size n X n row by row in the greedy way such that in no row or column or m X m square a symbol occurs more than once.

1, 2, 4, 4, 6, 9, 4, 6, 11, 16, 8, 7, 13, 18, 25, 8, 8, 13, 18, 27, 36, 8, 10, 13, 20, 29, 38, 49, 8, 10, 13, 20, 32, 38, 51, 64, 16, 13, 14, 22, 33, 40, 53, 66, 81, 16, 15, 14, 22, 33, 40, 56, 66, 83, 100, 16, 16, 15, 23, 33, 41, 57, 68, 85, 102

Offset: 1

Author

M. F. Hasler, Sep 20 2017

Keywords

• nonn
• tabl

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 m X m 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 m-1 (having up to (2m-1)^2 grid points). See A292673 for examples.

One can consider the infinite square array A(m,n) defined in the same way, but the lower triangular part of it is uninteresting: for m>n one has A(m,n) = n^2, i.e., the columns continue below the diagonal indefinitely with the same value, n^2.

Examples

			The infinite square array would look as follows: (but the sequence only lists the upper right triangle: 1; 2, 4; 4, 6, 9; 4, 6, 11, 16; ...):
   [1  2  4  4  8  8  8  8 16 ...]  m=1: A(1,n) = 2^ceil(log_2(n)) = A062383(n-1)
   [1\_4  6  6  7  8 10 10 13 ...]  m=2: A(2,n) = A292672(n)
   [1  4\_9 11 13 13 13 13 14 ...]  m=3: A(3,n) = A292673(n) : see here
   [1  4  9\16 18 18 20 20 22 ...]  m=4: A(4,n) = A292674(n) :  for examples
   [1  4  9 16\25 27 29 32 33 ...]  m=5: A(5,n) = A292675(n)
   [1  4  9 16 25\36 38 38 40 ...]  m=6: A(6,n) = A292676(n)
   [1  4  9 16 25 36\49 51 53 ...]  m=7: A(7,n) = A292677(n)
   [1  4  9 16 25 36 49\64 66 ...]  m=8: A(8,n) = A292678(n)
   [1  4  9 16 25 36 49 64\81 ...]  m=9: A(9,n) = A292679(n)
   [...   ...  ...  ...  ...  ...]
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..5050 (first 100 rows)
• Eric Weisstein's World of Mathematics, Moore Neighborhood.

Crossrefs

Cf. A292670, A292672, A292673, ..., A292679.

Programs

PARI

A(m, n, 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

def A(m, n): # change m for A292672, ..., A292679
    mx, S, N, b = 0, {1}, range(1, n+1), m # b is block size
    g = [[0 for j in range(n+b)] for i in range(n+b)]
    row, col = {i:set() for i in N}, {j:set() for j in N}
    offsets = [(i, j) for i in range(-b+1, 1) for j in range(-b+1, 1)]
    offsets += [(i, j) for i in range(-b+1, 0) for j in range(1, b)]
    for i in N:
        for j in N:
            rect = set(g[i+o[0]][j+o[1]] for o in offsets)
            e = min(S - row[i] - col[j] - rect)
            g[i][j] = e
            if e > mx:
                mx = e
                S.add(mx+1)
            row[i].add(e)
            col[j].add(e)
    return mx
print([A(m, n) for n in range(1, 12) for m in range(1, n+1)]) # Michael S. Branicky, Apr 13 2023