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%I #4 Feb 09 2018 17:36:08
%S 1,2,8,10,12,16,19,22,23,25,29,30,34,35,41,43,44,46,52,52,54,60,60,62,
%T 64,66,70,75,77,78,80,82,84,88,91,92,94,96,98,102,105,108,111,112,114,
%U 118,119,121,123,127,132,134,137,140,141,143,147,148,154,156
%N 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.
%C 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:
%C (1) 0 <= a(k) - a(k-1) <= 6 for k>=1;
%C (2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k-1) + d for infinitely many k.
%C ***
%C See A298000 and A297830 for guides to related sequences.
%H Clark Kimberling, <a href="/A297999/b297999.txt">Table of n, a(n) for n = 0..2000</a>
%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 8.
%e Complement: (b(n)) = (3,4,5,6,7,9,11,13,14,15,17, ...)
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[1]*b[n] - a[0]*b[n - 1] + n;
%t Table[{a[n], b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 3000}];
%t Table[a[n], {n, 0, 150}] (* A297999 *)
%t Table[b[n], {n, 0, 150}] (* A298110 *)
%t (* _Peter J. C. Moses_, Jan 16 2018 *)
%Y Cf. A297997, A298000, A297830, A298110.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Feb 09 2018
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