A065677 Maximal Diffy_length for quadruples of numbers <= n.

0, 4, 4, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Author

Jens Voß, Nov 13 2001

Keywords

• nonn

Comments

For quadruples of nonnegative integers a, b, c, d we let diffy([a, b, c, d]) := [|a-b|, |b-c|, |c-d|, |d-a|] (i.e. the quadruple of absolute differences of neighboring values, cyclically speaking) and Diffy_length([a, b, c, d]) := min { n in N | diffy^n([a, b, c, d]) = [0, 0, 0, 0] } (i.e. the minimum number of diffy iterations needed to convert [a, b, c, d] into [0, 0, 0, 0]).

Monotonically nondecreasing; the sequence A065678 (or A045794) is its "inverse" (i.e. A065678(n) = min {m | A065677(m) >= n})

Examples

			Diffy_length([0,0,0,1]) = 4 since diffy^4([0,0,0,1]) = diffy^3([0,0,1,1]) = diffy^2([0,1,0,1]) = diffy([1,1,1,1]) = [0,0,0,0], so A065677(1) >= 4 (considering all quadruples of numbers 0 and 1 shows that in fact A065677(1) = 4)
		

Links

• Raymond Greenwell, 73.35 The Game of Diffy, Math. Gazette, Vol. 73, No. 465 (Oct., 1989), pp. 222-225.
• Univ. Mass. Computer Science 121, The Diffy Game

Crossrefs

Cf. A065678 (or A045794).