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A298111 Solution b( ) of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*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.

%I A298111 #4 Feb 09 2018 11:10:24
%S A298111 3,4,5,6,7,8,9,11,12,14,15,17,18,20,21,23,24,25,26,28,30,31,32,33,35,
%T A298111 37,38,39,40,42,44,45,46,47,49,51,52,53,54,56,58,59,61,62,64,65,66,67,
%U A298111 69,70,71,73,75,76,78,79,81,82,83,84,86,87,88,90,92,93
%N A298111 Solution b( ) of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*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 A298111 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A298000.  See A297830 for a guide to related sequences.
%C A298111 Conjecture:  1/5 < a(n) - n*sqrt(2) < 3 for n >= 1.
%H A298111 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%t A298111 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A298111 a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 n;
%t A298111 j = 1; While[j < 80000, k = a[j] - j - 1;
%t A298111  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298111 u = Table[a[n], {n, 0, k}]; (* A298000 *)
%t A298111 v = Table[b[n], {n, 0, k}]; (* A298111 *)
%t A298111 Take[u, 50]
%t A298111 Take[v, 50]
%Y A298111 Cf. A297830, A298000.
%K A298111 nonn,easy
%O A298111 0,1
%A A298111 _Clark Kimberling_, Feb 09 2018