A292673 -id:A292673 - cp's OEIS frontend

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

M. F. Hasler, Sep 20 2017

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.

			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)
   [...   ...  ...  ...  ...  ...]

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

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

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.

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

A292672 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 2 X 2 block no symbol occurs twice.

Original entry on oeis.org

1, 4, 6, 6, 7, 8, 10, 10, 13, 15, 16, 17, 19, 20, 21, 22, 23, 25, 28, 30, 31, 32, 33, 35, 35, 37, 38, 39, 40, 41, 43, 44, 45, 47, 50, 52, 53, 55, 57, 58, 60, 60, 61, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 76, 78, 78, 79, 80, 82, 84, 85, 87, 89, 90, 92, 93, 94

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 2 X 2 rectangles have "floating" borders, so the constraint is actually equivalent to saying that any element must be different from all neighbors in a Moore neighborhood of range 1 (having up to 3*3=9 grid points).

			For n = 4, 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(4) = 6.
For n = 3 the result would be the upper 3 X 3 part of the above grid, showing that also a(3) = 6.

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..100 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Moore Neighborhood

Cf. A292670, A292671, A292673, ..., A292679.

a(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 X n board.

def a(n):
    m, s, N = 0, {1}, range(1, n+1)
    g = [[0 for j in range(n+2)] for i in range(n+2)]
    row, col = {i:set() for i in N}, {j:set() for j in N}
    for i in N:
        for j in N:
            rect = {g[i-1][j-1], g[i-1][j], g[i][j-1], g[i-1][j+1]}
            e = min(s - row[i] - col[j] - rect)
            g[i][j] = e
            row[i].add(e)
            col[j].add(e)
            if e > m:
                m = e
                s.add(m+1)
    return m
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Apr 13 2023

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

A292679 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 9 X 9 block no symbol occurs twice.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 83, 85, 88, 89, 91, 92, 94, 95, 95, 96, 97, 100, 102, 103, 104, 103, 105, 102, 103, 104, 104, 104, 104, 105, 107, 108, 108, 115, 114, 115, 111, 112, 112, 111, 113, 117, 118, 119, 120, 121, 122, 123, 124, 126, 126, 126, 126, 126, 126

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 9 X 9 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 8 (having up to 17*17 = 289 grid points).
See sequences A292672, A292673, A292674 for examples.

Eric Weisstein's World of Mathematics, Moore Neighborhood

Cf. A292670, A292671, A292672, ..., A292678.

a(n,m=9,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.

# uses function in A292673
print([A292673(n, b=9) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

A292674 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 4 X 4 block no symbol occurs twice.

Original entry on oeis.org

1, 4, 9, 16, 18, 18, 20, 20, 22, 22, 23, 23, 23, 24, 25, 26, 26, 26, 29, 32, 32, 34, 36, 38, 38, 38, 42, 42, 42, 44, 44, 45, 48, 49, 49, 49, 54, 54, 54, 56, 59, 59, 64, 65, 68, 69, 70, 73, 76, 78, 79, 79, 82, 82, 83, 86, 87, 89, 90, 92, 95, 95, 96, 96, 97, 97

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 4 X 4 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 3 (having up to 7*7 = 49 grid points).

			For n = 8, the grid is filled as follows:
  [ 1  2  3  4  5  6  7  8]
  [ 5  6  7  8  1  2  3  4]
  [ 9 10 11 12 13 14 15 16]
  [13 14 15 16  9 10 11 12]
  [ 2  3  4 17 18  5  6  7]
  [ 6  1  8  7  2  3  4 17]
  [10  5 12 19 20  1  8 13]
  [11  9 13 14 10 15 12 19]
whence a(8) = 20.

Andrew Howroyd, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Moore Neighborhood

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

a(n,m=4,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.

# uses function in A292673
print([A292673(n, b=4) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

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

A292675 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 5 X 5 block no symbol occurs twice.

Original entry on oeis.org

1, 4, 9, 16, 25, 27, 29, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 38, 40, 40, 40, 40, 40, 40, 40, 40, 41, 43, 42, 42, 46, 47, 48, 52, 53, 53, 54, 57, 58, 58, 59, 62, 63, 64, 66, 68, 70, 72, 73, 74, 75, 75, 78, 78, 79, 80, 82, 83, 85, 86, 88, 91, 92, 97, 96, 98

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 5 X 5 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 4 (having up to 9*9 = 81 grid points).

Andrew Howroyd, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Moore Neighborhood

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

a(n,m=5,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.

# uses function in A292673
print([A292673(n, b=5) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

Terms a(55) and beyond from Andrew Howroyd, Feb 22 2020

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.

Original entry on oeis.org

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

M. F. Hasler, Sep 20 2017

nonn

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.

Eric Weisstein's World of Mathematics, Moore Neighborhood

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

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.

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

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

A292676 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 6 X 6 block no symbol occurs twice.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 38, 38, 40, 40, 41, 41, 43, 45, 48, 48, 50, 49, 49, 49, 49, 50, 50, 51, 51, 52, 51, 52, 53, 53, 53, 53, 53, 53, 55, 53, 55, 55, 59, 59, 59, 61, 65, 64, 66, 70, 68, 69, 72, 73, 78, 78, 79, 84, 85, 85, 86, 90, 90, 90, 94, 93, 96, 97, 99, 102, 105, 106, 106, 107

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 6 X 6 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 5 (having up to 11*11 = 121 grid points).

Eric Weisstein's World of Mathematics, Moore Neighborhood

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

a(n,m=6,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.

# uses function in A292673
print([A292673(n, b=6) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

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

A292677 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 7 X 7 square any symbol occurs twice.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 51, 53, 56, 57, 59, 60, 60, 61, 62, 64, 66, 65, 64, 62, 62, 64, 66, 65, 67, 67, 67, 66, 67, 69, 67, 69, 69, 70, 70, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 75, 76, 78, 80, 80, 83, 82, 83, 87, 94, 99, 106, 107, 108, 109, 110, 111, 112, 112

Offset: 1

M. F. Hasler, Sep 20 2017

nonn

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 7 X 7 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 6 (having up to 13*13 = 169 grid points).

Eric Weisstein's World of Mathematics, Moore Neighborhood

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

a(n,m=7,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.

# uses function in A292673
print([A292673(n, b=7) for n in range(1, 101)]) # Michael S. Branicky, Apr 13 2023

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

cp's OEIS Frontend

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.

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A292672 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 2 X 2 block no symbol occurs twice.

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A292679 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 9 X 9 block no symbol occurs twice.

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A292674 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 4 X 4 block no symbol occurs twice.

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A292675 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 5 X 5 block no symbol occurs twice.

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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.

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A292676 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 6 X 6 block no symbol occurs twice.

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A292677 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 7 X 7 square any symbol occurs twice.