A297999 Solution (a(n)) of the near-complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, , b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 8, 10, 12, 16, 19, 22, 23, 25, 29, 30, 34, 35, 41, 43, 44, 46, 52, 52, 54, 60, 60, 62, 64, 66, 70, 75, 77, 78, 80, 82, 84, 88, 91, 92, 94, 96, 98, 102, 105, 108, 111, 112, 114, 118, 119, 121, 123, 127, 132, 134, 137, 140, 141, 143, 147, 148, 154, 156

Offset

0,2

Comments

The sequence (a(n)) generated by the equation a(n) = a(1)*b(n) - a(0)*b(n-1) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 52. If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures:

(1) 0 <= a(k) - a(k-1) <= 6 for k>=1;

(2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k-1) + d for infinitely many k.

***

See A298000 and A297830 for guides to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..2000

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 8.
Complement: (b(n)) = (3,4,5,6,7,9,11,13,14,15,17, ...)

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[1]*b[n] - a[0]*b[n - 1] + n;
Table[{a[n], b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 3000}];
Table[a[n], {n, 0, 150}]  (* A297999 *)
Table[b[n], {n, 0, 150}]  (* A298110 *)
(* Peter J. C. Moses, Jan 16 2018 *)

Crossrefs

Cf. A297997, A298000, A297830, A298110.

Sequence in context: A022298 A229090 A265670 * A287088 A296342 A328559

Adjacent sequences: A297996 A297997 A297998 * A298000 A298001 A298002

Keyword

nonn,easy

Author

Clark Kimberling, Feb 09 2018

Status

approved