A127208 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).

1, 3, 4, 7, 11, 15, 18, 21, 26, 29, 31, 39, 47, 51, 57, 63, 71, 76, 99, 113, 120, 123, 127, 131, 191, 199, 223, 239, 241, 247, 255, 322, 367, 439, 443, 475, 493, 502, 511, 521, 708, 815, 843, 863, 943, 983, 1003, 1013, 1023, 1364, 1365, 1499, 1695, 1871, 1959

Offset

1,2

Comments

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.

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(A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755,...)

Mathematica

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

Crossrefs

Cf. A227885.

Sequence in context: A023563 A050120 A039010 * A027022 A120365 A166375

Adjacent sequences: A127205 A127206 A127207 * A127209 A127210 A127211

Keyword

nonn

Author

T. D. Noe, Jan 09 2007

Status

approved