A124168 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).

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, 120, 125, 127, 128, 144, 149, 208, 233, 236, 248, 253, 255, 256, 274, 377, 401, 464, 492, 504, 509, 511, 512, 610, 773, 912, 927, 976, 987, 1004, 1016, 1021, 1023, 1024

Offset

1,2

Comments

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)...

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

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

Formula

Union(A000045, A000073, A000078, A001591, A001592, ...)

Mathematica

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]]]
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 *)

Crossrefs

Cf. A000045, A000073, A000078, A001591, A001592, A124257.

Cf. A227880 (primes here).

Sequence in context: A247350 A057484 A091997 * A309708 A285929 A309880

Adjacent sequences: A124165 A124166 A124167 * A124169 A124170 A124171

Keyword

nonn

Author

Carlos Alves, Dec 03 2006

Extensions

Edited by N. J. A. Sloane, Dec 15 2006

Status

approved