A393159 Square array A(n, k), n, k >= 0, read by antidiagonals: A(n, k) is the largest possible integer Euclidean distance between two lattice points with X-coordinates in the interval 0..n and Y-coordinates in the interval 0..k.

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 5, 5, 5, 6, 7, 8, 7, 6, 5, 5, 5, 6, 7, 8, 9, 8, 7, 6, 5, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12

Offset

0,4

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Rémy Sigrist, Colored representation of the array for n, k < 1000

Rémy Sigrist, PARI program

Index entries related to Pythagorean Triples

Formula

A(n, k) = A(k, n).

A(n, k) >= max(n, k).

A(n, 0) = n.

If n^2 + k^2 = v^2 for some nonnegative integer v, then A(n, k) = v, otherwise A(n, k) = max(A(n-1, k), A(n, k-1)).

Example

Square array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12
    1 |  1   1   2   3   4   5   6   7   8   9  10  11  12
    2 |  2   2   2   3   4   5   6   7   8   9  10  11  12
    3 |  3   3   3   3   5   5   6   7   8   9  10  11  12
    4 |  4   4   4   5   5   5   6   7   8   9  10  11  12
    5 |  5   5   5   5   5   5   6   7   8   9  10  11  13
    6 |  6   6   6   6   6   6   6   7  10  10  10  11  13
    7 |  7   7   7   7   7   7   7   7  10  10  10  11  13
    8 |  8   8   8   8   8   8  10  10  10  10  10  11  13
    9 |  9   9   9   9   9   9  10  10  10  10  10  11  15
   10 | 10  10  10  10  10  10  10  10  10  10  10  11  15
   11 | 11  11  11  11  11  11  11  11  11  11  11  11  15
   12 | 12  12  12  12  12  13  13  13  13  15  15  15  15

Prog

(PARI) \\ See Links section.
(Python)
from functools import lru_cache
from math import isqrt
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A393159_A(n, k): return n if k==0 else A393159_A(k, n) if k>n else isqrt(v) if is_square(v:=n**2+k**2) else max(A393159_A(n-1, k), A393159_A(n, k-1)) # Chai Wah Wu, Feb 08 2026

Crossrefs

Cf. A003984, A393160 (main diagonal).

Sequence in context: A231205 A003984 A087061 * A344838 A344835 A082860

Adjacent sequences: A393156 A393157 A393158 * A393160 A393161 A393162

Keyword

nonn,tabl

Author

Rémy Sigrist, Feb 03 2026

Status

approved