A065678 Minimum value t such that all quadruples of Diffy_length >= n have a maximal value >= t.

0, 1, 1, 1, 1, 3, 3, 4, 9, 11, 13, 31, 37, 44, 105, 125, 149, 355, 423, 504, 1201, 1431, 1705, 4063, 4841, 5768, 13745, 16377, 19513, 46499, 55403, 66012, 157305, 187427, 223317, 532159, 634061, 755476, 1800281, 2145013, 2555757, 6090307, 7256527

Offset: 0

Author

Jens Voß, Nov 13 2001

Keywords

• nonn
• easy

Comments

Another version of A045794, which has further information including a formula.

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]).

The "inverse" of sequence A065677 (i.e. A065678(n) = min {m | A065677(m) >= n})

Examples

			Since Diffy_length([0,0,0,0]) = 0 and Diffy_length([0,0,0,1]) = 4, we have A065678(1) = A065678(2) = A065678(3) = A065678(4) = 1.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• A. Behn, C. Kribs-Zaleta and V. Ponomarenko, The convergence of difference boxes, Amer. Math. Monthly 112 (2005), no. 5, 426-439.
• J. Copeland and J. Haemer, Work: Differences Among Women, SunExpert, 1999, pp. 38-43.
• Raymond Greenwell, The Game of Diffy, Math. Gazette, Oct 1989, p. 222.
• Peter J. Kernan (pete(AT)theory2.phys.cwru.edu), Algorithm and code [Broken link]
• Dawn J. Lawrie, The Diffy game.
• Univ. Mass. Computer Science 121, The Diffy Game [Broken link]
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,1,0,0,1).

Crossrefs

Cf. A065677.

Programs

PARI

concat(0, Vec(x*(x-1)*(x^8+x^6+x^5+x^4+x^3+3*x^2+2*x+1)/(x^9+x^6+3*x^3-1) + O(x^100))) \\ Colin Barker, Feb 18 2015

Formula

From Colin Barker, Feb 18 2015: (Start)

a(n) = 3*a(n-3)+a(n-6)+a(n-9).

G.f.: x*(x-1)*(x^8+x^6+x^5+x^4+x^3+3*x^2+2*x+1) / (x^9+x^6+3*x^3-1).

(End)