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

1,4

Comments

Another version of A065678, which has further information.

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).

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)

Example

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).

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 *)

Prog

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}
(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

Crossrefs

Sequence in context: A128036 A332311 A332340 * A065678 A022598 A107635

Adjacent sequences: A045791 A045792 A045793 * A045795 A045796 A045797

Keyword

nonn,nice,easy

Author

Ikuo Kiyokawa (kiyo19(AT)mxr.meshnet.or.jp)

Extensions

Reference and better description from Erich Friedman

Status

approved