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%I #16 Jun 13 2015 00:50:30
%S 0,1,1,1,1,3,3,4,9,11,13,31,37,44,105,125,149,355,423,504,1201,1431,
%T 1705,4063,4841,5768,13745,16377,19513,46499,55403,66012,157305,
%U 187427,223317,532159,634061,755476,1800281,2145013,2555757,6090307,7256527
%N Minimum value t such that all quadruples of Diffy_length >= n have a maximal value >= t.
%C Another version of A045794, which has further information including a formula.
%C 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]).
%C The "inverse" of sequence A065677 (i.e. A065678(n) = min {m | A065677(m) >= n})
%H Colin Barker, <a href="/A065678/b065678.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Behn, C. Kribs-Zaleta and V. Ponomarenko, <a href="http://www.jstor.org/stable/30037493">The convergence of difference boxes</a>, Amer. Math. Monthly 112 (2005), no. 5, 426-439.
%H J. Copeland and J. Haemer, <a href="http://alumnus.caltech.edu/~copeland/work/PDF/1999-02-women.pdf">Work: Differences Among Women</a>, SunExpert, 1999, pp. 38-43.
%H Raymond Greenwell, <a href="http://www.jstor.org/stable/3618447">The Game of Diffy</a>, Math. Gazette, Oct 1989, p. 222.
%H Peter J. Kernan (pete(AT)theory2.phys.cwru.edu), <a href="http://theory2.phys.cwru.edu/~pete/sequence.html">Algorithm and code</a> [Broken link]
%H Dawn J. Lawrie, <a href="http://www.cs.loyola.edu/~lawrie/CS630/F03/Projects/PA1/index.html">The Diffy game</a>.
%H Univ. Mass. Computer Science 121, <a href="http://www-unix.oit.umass.edu/~cs121/projects/project3/p3.htm">The Diffy Game</a> [Broken link]
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,1,0,0,1).
%F From _Colin Barker_, Feb 18 2015: (Start)
%F a(n) = 3*a(n-3)+a(n-6)+a(n-9).
%F 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).
%F (End)
%e 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.
%o (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
%Y Cf. A065677.
%K nonn,easy
%O 0,6
%A _Jens Voß_, Nov 13 2001
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