A297827 -id:A297827 - cp's OEIS frontend

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

Clark Kimberling, Feb 04 2018

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.

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

Cf. A297826, A297827.

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 *)

A297829 Duplicates in A297826.

Original entry on oeis.org

51, 59, 243, 279, 287, 295, 363, 371, 422, 430, 538, 587, 684, 872, 934, 1075, 1083, 1091, 1232, 1268, 1304, 1312, 1320, 1388, 1396, 1515, 1598, 1634, 1642, 1650, 1718, 1726, 1855, 1891, 1899, 1907, 1975, 1983, 2034, 2042, 2093, 2101, 2209, 2258, 2355, 2363

Offset: 1

Clark Kimberling, Feb 04 2018

Cf. A297826.

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]];
du = Differences[Table[a[n], {n, 0, 1000}]]  (* A297827 *)
w = Flatten[Position[du, 0]]; Map[a, w] (* A297829 *)

cp's OEIS Frontend

A297828 Difference sequence of A297997.

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