A330190 Symmetric matrix read by antidiagonals: f(i,j) = 1 + gcd(f(i-1,j), f(i,j-1)), where f(1,j) and f(i,1) are 1.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 2, 4, 4, 2, 2, 1, 1, 2, 3, 3, 5, 3, 3, 2, 1, 1, 2, 2, 4, 2, 2, 4, 2, 2, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 2, 3, 3, 5, 5, 5, 5, 5, 3, 3, 2, 1

Offset: 1

Author

Nathaniel J. Strout, Dec 04 2019

Keywords

• nonn
• tabl
• look

Comments

This matrix when displayed in a gray scale, from least to greatest, forms spikes of increasing numbers because large sections of the antidiagonals are the same number. See examples section.

Examples

			An example of a triangle described in the comment:
  ...........
  ...........
  ..........2
  ........2 3
  ......2 3 4
  ....2 3 4 5
Array begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
  1, 2, 3, 2, 3, 2, 3, 2, 3, 2, ...
  1, 2, 2, 3, 4, 3, 4, 3, 4, 3, ...
  1, 2, 3, 4, 5, 2, 3, 4, 5, 2, ...
  1, 2, 2, 3, 2, 3, 4, 5, 6, 3, ...
  1, 2, 3, 4, 3, 4, 5, 6, 7, 2, ...
  1, 2, 2, 3, 4, 5, 6, 7, 8, 3, ...
  1, 2, 3, 4, 5, 6, 7, 8, 9, 4, ...
  1, 2, 2, 3, 2, 3, 2, 3, 4, 5, ...
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows n = 1..150, flattened)
• Nathaniel J. Strout, 1000 X 1000 grid
• Michael De Vlieger, 2048 X 2048 grid with color function where black = 1, red = 2 and magenta represents the maximum value in the grid (i.e., f(312,768) = f(768,312) = 41).

Programs

Mathematica

f[1, j_] := f[1, j] = 1; f[i_, 1] := f[i, 1] = 1; f[i_, j_] := f[i, j] = 1 + GCD[f[i - 1, j], f[i, j - 1]]; Table[f[m - k + 1, k], {m, 13}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, Aug 03 2022 *)

PARI

T(n)={my(M=matrix(n,n,i,j,1)); for(i=2, n, for(j=2, n, M[i,j] = 1 + gcd(M[i-1,j], M[i,j-1]))); M}
{ my(A=T(10)); for(i=1, #A, print(A[i,])) } \\ Andrew Howroyd, Jan 25 2020