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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 14 01:17 EDT 2021. Contains 342941 sequences. (Running on oeis4.)