

A297826


Solution (a(n)) of the nearcomplementary equation a(n) = a(1)*b(n1)  a(0)*b(n2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.


17



1, 2, 7, 9, 11, 15, 18, 21, 22, 24, 28, 29, 33, 34, 40, 42, 43, 45, 51, 51, 53, 59, 59, 61, 63, 65, 69, 74, 76, 77, 79, 81, 83, 87, 90, 91, 93, 95, 97, 101, 104, 107, 110, 111, 113, 117, 118, 120, 122, 126, 131, 133, 136, 139, 140, 142, 146, 147, 153, 155
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OFFSET

0,2


COMMENTS

The sequence (a(n)) generated by the equation a(n) = a(1)*b(n1)  a(0)*b(n2) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 51. If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures:
(1) 0 <= a(k)  a(k1) <= 6 for k>=1;
(2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k1) + d for infinitely many k.
***
See A297830 for a guide to related sequences.


LINKS



EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 7.
Complement: (b(n)) = (3, 4, 5, 6, 8,10,12,13,14,16, ...)


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]];
Table[a[n], {n, 0, 300}] (* A297826 *)
Table[b[n], {n, 0, 300}] (* A297997 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



