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%I #5 Feb 09 2018 17:36:18
%S 3,4,5,6,7,9,11,13,14,15,17,18,20,21,24,26,27,28,31,32,33,36,37,38,39,
%T 40,42,45,47,48,49,50,51,53,55,56,57,58,59,61,63,65,67,68,69,71,72,73,
%U 74,76,79,81,83,85,86,87,89,90,93,95,97,99,100,101,103
%N Solution (b(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-1) - a(0)*b(n-2) + 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:
%C (1) 1 <= b(k) - b(k-1) <= 3 for k>=1;
%C (2) if d is in {1,2,3}, then b(k) = b(k-1) + d for infinitely many k.
%C ***
%C See A298000 and A297830 for guides to related sequences.
%H Clark Kimberling, <a href="/A298110/b298110.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: A298110 = (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. A297999, A298000, A297830.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Feb 09 2018
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