A297828 Difference sequence of A297997.

1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1

Offset: 1

Author

Clark Kimberling, Feb 04 2018

Keywords

• nonn
• easy

Comments

Conjectures:

(1) 2 <= a(k) <= 4 for k>=1;

(2) if d is in {1,2,3}, then a(k) = d for infinitely many k.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A297826, A297827.

Programs

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n;
b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]];
u = Table[a[n], {n, 0, 300}](* A297826 *)
v = Table[b[n], {n, 0, 300}](* A297997 *)
Differences[u];  (* A297827 *)
Differences[v];  (* A297828 *)
(* Peter J. C. Moses, Jan 03 2017 *)