A199408 Triangle T(n,k) = n + k - gcd(n,k) read by rows, 1 <= n, 1 <= k <= n.

1, 2, 2, 3, 4, 3, 4, 4, 6, 4, 5, 6, 7, 8, 5, 6, 6, 6, 8, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 8, 10, 8, 12, 12, 14, 8, 9, 10, 9, 12, 13, 12, 15, 16, 9, 10, 10, 12, 12, 10, 14, 16, 16, 18, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 12, 12, 12, 12, 16, 12

Offset: 1

Author

Brian Hopkins, Nov 05 2011

Keywords

• nonn
• tabl
• easy

Comments

A diagonal of an n by k rectangle drawn on a square grid passes through T(n,k) squares: the diagonal enters n squares crossing horizontal segments and enters k squares crossing vertical segments. Gcd(n,k) counts the squares entered at a lattice point, which have been over-counted.

Examples

			T(6,4) = 6 + 4 - 2 = 8.
Triangular array begins
  1
  2  2
  3  4  3
  4  4  6  4
  5  6  7  8  5
  6  6  6  8 10  6
  7  8  9 10 11 12  7
  8  8 10  8 12 12 14  8
		

References

• M. Ollerton, Mathematics Teacher's Handbook, Continuum, 2009, pp. 14-15.

Links

• Seiichi Manyama, Rows n = 1..140, flattened
• Association of Teachers of Mathematics, Points of Departure 1, Derby, 1972.

Crossrefs

Cf. A049627, A074712. Third column A061800.

Programs

PARI

T(n,k) = n + k - gcd(n,k); \\ Michel Marcus, Aug 04 2018

Formula

T(d*a,d*b) = d*T(a,b).