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%I #24 Jun 13 2015 00:49:49
%S 1,1,1,3,3,4,9,11,13,31,37,44,105,125,149,355,423,504,1201,1431,1705,
%T 4063,4841,5768,13745,16377,19513,46499,55403,66012,157305,187427,
%U 223317,532159,634061,755476,1800281,2145013,2555757,6090307,7256527
%N Consider all quadruples {a,b,c,d} which reach {k,k,k,k} in n steps under map {a,b,c,d}->{|a-b|,|b-c|,|c-d|,|d-a|}; look at max{a,b,c,d}; sequence gives minimal value of this.
%C Another version of A065678, which has further information.
%H Colin Barker, <a href="/A045794/b045794.txt">Table of n, a(n) for n = 1..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 Equals [ b(0)+b(2), b(1)+b(2), b(3), b(2)+b(4), b(3)+b(4), b(5), ... ], where b() = A000073. - Peter J. Kernan (pete(AT)theory2.phys.cwru.edu).
%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^7-x^6+x^5+x^2+x+1) / (x^9+x^6+3*x^3-1).
%F (End)
%e a(7) = 9 because {0,1,4,9}->{1,3,5,9}->{2,2,4,8}->{0,2,4,6}->{2,2,2,6}->{0,0,4,4}->{0,4,0,4}->{4,4,4,4} (7 steps and no quadruple with a,b,c,d <= 8 works).
%t LinearRecurrence[{0,0,3,0,0,1,0,0,1},{1,1,1,3,3,4,9,11,13},50] (* _Harvey P. Dale_, May 30 2015 *)
%o a=(0 1 0 1); while 1{s=a_3-a_2-a_1; s%2?a*=2:s/=2; a+=s; a_2=a_0+a_1; a_0=0; a_1=s; print a_3}
%o (PARI) Vec(-x*(x^7-x^6+x^5+x^2+x+1)/(x^9+x^6+3*x^3-1) + O(x^100)) \\ _Colin Barker_, Feb 18 2015
%K nonn,nice,easy
%O 1,4
%A Ikuo Kiyokawa (kiyo19(AT)mxr.meshnet.or.jp)
%E Reference and better description from _Erich Friedman_
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