A045794 -id:A045794 - cp's OEIS frontend

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

Jens Voß, Nov 13 2001

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})

			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.

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).

Cf. A065677.

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

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)

A065677 Maximal Diffy_length for quadruples of numbers <= n.

Original entry on oeis.org

0, 4, 4, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Jens Voß, Nov 13 2001

nonn

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]).

Monotonically nondecreasing; the sequence A065678 (or A045794) is its "inverse" (i.e. A065678(n) = min {m | A065677(m) >= n})

			Diffy_length([0,0,0,1]) = 4 since diffy^4([0,0,0,1]) = diffy^3([0,0,1,1]) = diffy^2([0,1,0,1]) = diffy([1,1,1,1]) = [0,0,0,0], so A065677(1) >= 4 (considering all quadruples of numbers 0 and 1 shows that in fact A065677(1) = 4)

Raymond Greenwell, 73.35 The Game of Diffy, Math. Gazette, Vol. 73, No. 465 (Oct., 1989), pp. 222-225.
Univ. Mass. Computer Science 121, The Diffy Game

Cf. A065678 (or A045794).

A034803 Consider the sequence of 4-tuples {0,a,b,c} (c>=a+b; a,b,c>0) which have the smallest integer 'c' required to reach {k,k,k,k} in n steps under map {r,s,t,u}->{|r-s|,|s-t|,|t-u|,|u-r|}. This sequence gives the second term 'a' of these quadruples.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 2, 2, 5, 6, 7, 17, 20, 24, 57, 68, 81, 193, 230, 274, 653, 778, 927, 2209, 2632, 3136, 7473, 8904, 10609, 25281, 30122, 35890, 85525, 101902, 121415, 289329, 344732, 410744, 978793, 1166220, 1389537, 3311233, 3945294, 4700770

Offset: 1

N. J. A. Sloane

nonn

			a(10)=5 because {0, 5, 14, 31}->{5, 9, 17, 31}->{4, 8, 14, 26}->{4, 6, 12, 22}->{2, 6, 10, 18}->{4, 4, 8, 16}->{0, 4, 8, 12}->{4, 4, 4, 12}->{0, 0, 8, 8}->{0, 8, 0, 8}->{8, 8, 8, 8} ('a'=5 in the first 4-tuple and there is no quadruple with a+b<=c <= 31 and 10 steps).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,1,0,0,1).

A034804, A045794 (or A065678) give the terms 'b' and 'c' respectively.

LinearRecurrence[{0,0,3,0,0,1,0,0,1},{1,0,0,1,0,1,1,2,2},60] (* Harvey P. Dale, Jun 13 2017 *)

a(n)= Trib(2*q-3)+Trib(2*q-1) if r=0; Trib(2*q-2)+Trib(2*q-1) if r=1; Trib(2*q) if r=2 where q=[(n-1)/3], r=n-1 (mod 3) and Trib is the tribonacci sequence (A000073) with Trib(-3)=0, Trib(-2)=-1, Trib(-1)=1. G.f.: (x^8-2*x^7+3*x^6-x^5+2*x^3-1)/(x^9+x^6+3*x^3-1). Recurrence: a(n)=3*a(n-3)+a(n-6)+a(n-9), n >= 10.

Better description, more terms, formula, etc. from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 24 2001

Minor edits from Michael B. Porter, Feb 03 2010

A034804 Consider the sequence of 4-tuples {0,a,b,c} (c>=a+b; a,b,c>0) which have the smallest integer 'c' required to reach {k,k,k,k} in n steps under map {r,s,t,u}->{|r-s|,|s-t|,|t-u|,|u-r|}. This sequence gives the third term 'b' of these quadruples.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 4, 5, 6, 14, 17, 20, 48, 57, 68, 162, 193, 230, 548, 653, 778, 1854, 2209, 2632, 6272, 7473, 8904, 21218, 25281, 30122, 71780, 85525, 101902, 242830, 289329, 344732, 821488, 978793, 1166220, 2779074, 3311233, 3945294, 9401540

Offset: 1

N. J. A. Sloane

nonn

			a(10)=14 because {0, 5, 14, 31}->{5, 9, 17, 31}->{4, 8, 14, 26}->{4, 6, 12, 22}->{2, 6, 10, 18}->{4, 4, 8, 16}->{0, 4, 8, 12}->{4, 4, 4, 12}->{0, 0, 8, 8}->{0, 8, 0, 8}->{8, 8, 8, 8} ('b'=14 in the first 4-tuple and there is no quadruple with a+b<=c<=31 and 10 steps).

A034803, A045794 (or A065678) give the terms 'a' and 'c' respectively.

a(n)= 2*Trib(2*q) if r=0; Trib(2*q-1)+Trib(2*q+1) if r=1; Trib(2*q)+Trib(2*q+1) if r=2 where q=[(n-1)/3], r=n-1 (mod 3) and Trib denotes the tribonacci sequence (A000073) with Trib(-1)=1. G.f.: (-x^7+2*x^6-2*x^5+2*x^4-2*x^3-x)/(x^9+x^6+3*x^3-1). Recurrence: a(n)=3*a(n-3)+a(n-6)+a(n-9), n >= 10.

Better description, more terms, formula, etc. from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 24 2001

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

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A065677 Maximal Diffy_length for quadruples of numbers <= n.

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A034803 Consider the sequence of 4-tuples {0,a,b,c} (c>=a+b; a,b,c>0) which have the smallest integer 'c' required to reach {k,k,k,k} in n steps under map {r,s,t,u}->{|r-s|,|s-t|,|t-u|,|u-r|}. This sequence gives the second term 'a' of these quadruples.

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A034804 Consider the sequence of 4-tuples {0,a,b,c} (c>=a+b; a,b,c>0) which have the smallest integer 'c' required to reach {k,k,k,k} in n steps under map {r,s,t,u}->{|r-s|,|s-t|,|t-u|,|u-r|}. This sequence gives the third term 'b' of these quadruples.

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Keywords

Examples

Crossrefs

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Extensions