A226246 Triangle T(n, k), read by rows 1<=n, 1<=k<=n: Number of cells touched by a unit-width diagonal in a regular n X k grid

1, 2, 4, 3, 6, 7, 4, 8, 10, 10, 5, 8, 11, 14, 13, 6, 10, 14, 14, 16, 16, 7, 12, 13, 16, 19, 20, 19, 8, 14, 16, 20, 20, 22, 24, 22, 9, 14, 17, 18, 21, 22, 25, 28, 25, 10, 16, 20, 20, 26, 24, 28, 30, 30, 28, 11, 18, 21, 22, 25, 26, 29, 30, 33, 34, 31

Offset

1,2

References

J. D. E. Konhauser, D. J. Velleman and S. Wagon, Which Way Did the Bicycle Go?, Cambridge University Press, 1996, page 179.

Links

Andrew Woods, Table of n, a(n) for n = 1..5050

Formula

Let g := gcd(n,k), r := sqrt(n*n+k*k)/2.

T(n,k) = n+k+g+2*(g*floor(r/g)-floor(r/min(n,k))-1)

Example

A paintbrush of unit width is dragged centrally along the diagonal of a rectangular 5 X 7 grid. The number of squares in the grid which contain paint in their interiors is T(5,7) = 19.

Crossrefs

The zero-width case is A199408 (or A074712).

Sequence in context: A242424 A232271 A194277 * A216623 A297551 A297673

Adjacent sequences: A226243 A226244 A226245 * A226247 A226248 A226249

Keyword

nonn,tabl

Author

Andrew Woods, Jun 01 2013

Status

approved