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%I #18 Mar 16 2023 04:53:20
%S 1,2,3,4,5,7,8,13,15,16,21,24,29,31,32,34,44,55,56,61,63,64,81,89,108,
%T 120,125,127,128,144,149,208,233,236,248,253,255,256,274,377,401,464,
%U 492,504,509,511,512,610,773,912,927,976,987,1004,1016,1021,1023,1024
%N Union of all n-Fibonacci sequences, that is, all sequences s(0) = s(1) = ... = s(n-2) = 0, s(n-1) = 1 and for k >= n, s(k) = s(k-1) + ... + s(k-n).
%C Note that an n-Fibonacci sequence contains the numbers 2^k numbers for k<n. We also get 2^n-1, 2^(n+1)-3, 2^(n+2)-8, ... The sequence -1, -3, -8, continues following A001792 (for n large)...
%C Noe and Post conjectured that the only positive terms that are common to any two distinct n-step Fibonacci sequences are the powers of 2 that begin each sequence and 13 (in 2- and 3-step) and 504 (in 3- and 7-step). Perhaps we should also include 8 (in 2- and 4-step). - _T. D. Noe_, Dec 05 2006
%H T. D. Noe, <a href="/A124168/b124168.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.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
%F Union(A000045, A000073, A000078, A001591, A001592, ...)
%t NFib25[nfb_] := Transpose[NestList[Join[Drop[ #, {1}], {Plus @@ #}] &, Map[If[ # == nfb, 1, 0] &, Range[nfb]], 25]][[ -1]]; Union[Flatten[Map[NFib25, Range[2, 20]]]][[Range[100]]]
%t NFib[nfb_, lim_] := Module[{f = 2^Range[0, nfb - 1]}, While[f[[-1]] <= lim, AppendTo[f, Total[Take[f, -nfb]]]]; Most[f]]; lim = 12; Union[Flatten[Table[NFib[i, 2^lim], {i, 2, lim + 1}]]] (* _T. D. Noe_, Oct 25 2013 *)
%Y Cf. A000045, A000073, A000078, A001591, A001592, A124257.
%Y Cf. A227880 (primes here).
%K nonn
%O 1,2
%A _Carlos Alves_, Dec 03 2006
%E Edited by _N. J. A. Sloane_, Dec 15 2006
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