A127232 -id:A127232 - cp's OEIS frontend

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

T. D. Noe, Jan 09 2007

nonn

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.

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

Cf. A227885.

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

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

cp's OEIS Frontend

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).

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