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A298003 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*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.

%I A298003 #4 Feb 08 2018 21:49:05
%S A298003 3,4,5,6,7,8,10,11,13,14,16,17,19,20,22,23,24,25,27,29,30,31,32,34,36,
%T A298003 37,38,39,41,43,44,45,46,48,50,51,52,53,55,57,58,60,61,63,64,65,66,68,
%U A298003 69,70,72,74,75,77,78,80,81,82,83,85,86,87,89,91,92,94
%N A298003 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*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.
%C A298003 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A297830, which includes a guide to related sequences.
%C A298003 Conjecture:  3/5 < a(n) - n*sqrt(2)*n < 3 for n >= 1.
%H A298003 Clark Kimberling, <a href="/A298003/b298003.txt">Table of n, a(n) for n = 0..10000</a>
%H A298003 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 A298003 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A298003 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n;
%t A298003 j = 1; While[j < 80000, k = a[j] - j - 1;
%t A298003 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298003 u = Table[a[n], {n, 0, k}]; (* A297830 *)
%t A298003 v = Table[b[n], {n, 0, k}]; (* A298003 *)
%t A298003 Take[u, 50]
%t A298003 Take[v, 50]
%Y A298003 Cf. A297830.
%K A298003 nonn,easy
%O A298003 0,1
%A A298003 _Clark Kimberling_, Feb 08 2018