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%I #16 Dec 30 2023 16:54:07
%S 1,0,0,1,0,1,1,2,2,5,6,7,17,20,24,57,68,81,193,230,274,653,778,927,
%T 2209,2632,3136,7473,8904,10609,25281,30122,35890,85525,101902,121415,
%U 289329,344732,410744,978793,1166220,1389537,3311233,3945294,4700770
%N 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.
%H Harvey P. Dale, <a href="/A034803/b034803.txt">Table of n, a(n) for n = 1..1000</a>
%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 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.
%e 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).
%t 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 *)
%Y A034804, A045794 (or A065678) give the terms 'b' and 'c' respectively.
%K nonn
%O 1,8
%A _N. J. A. Sloane_
%E Better description, more terms, formula, etc. from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 24 2001
%E Minor edits from _Michael B. Porter_, Feb 03 2010
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