A045794 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.
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: 1
Examples
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).
Links
- Colin Barker, Table of n, a(n) for n = 1..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).
Programs
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Mathematica
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 *) -
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
Formula
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).
From Colin Barker, Feb 18 2015: (Start)
a(n) = 3*a(n-3)+a(n-6)+a(n-9).
G.f.: -x*(x^7-x^6+x^5+x^2+x+1) / (x^9+x^6+3*x^3-1).
(End)
Extensions
Reference and better description from Erich Friedman
Comments