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%I #7 Oct 27 2013 13:51:37
%S 1,3,4,7,11,15,18,21,26,29,31,39,47,51,57,63,71,76,99,113,120,123,127,
%T 131,191,199,223,239,241,247,255,322,367,439,443,475,493,502,511,521,
%U 708,815,843,863,943,983,1003,1013,1023,1364,1365,1499,1695,1871,1959
%N Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).
%C Noe and Post conjectured that the only positive terms that are common to any two distinct n-step Lucas sequences are the Mersenne numbers (A001348) that begin each sequence and 7 and 11 (in 2- and 3-step) and 5071 (in 3- and 4-step). The intersection of this sequence with the union of all the n-step Fibonacci sequences (A124168) appears to consist of 4, 21, 29, the Mersenne numbers 2^n-1 for all n and the infinite set of Eulerian numbers in A127232.
%H T. D. Noe, <a href="/A127208/b127208.txt">Table of n, a(n) for n=1..1000</a>
%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.pdf">Primes in Fibonacci n-step and Lucas n-step Sequences</a>, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
%F Union(A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755,...)
%t LucasSequence[n_,kMax_] := Module[{a,s,lst={}}, a=Join[Table[ -1,{n-1}],{n}]; While[s=Plus@@a; a=RotateLeft[a]; a[[n]]=s; s<=kMax, AppendTo[lst,s]]; lst]; nn=10; t={}; Do[t=Union[t,LucasSequence[n,2^(nn+1)]], {n,2,nn}]; t
%Y Cf. A227885.
%K nonn
%O 1,2
%A _T. D. Noe_, Jan 09 2007
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