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.
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
Keywords
Examples
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).
Links
- 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).
Programs
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Mathematica
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 *)
Formula
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.
Extensions
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