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A298006 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, 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 A298006 #4 Feb 09 2018 11:09:56
%S A298006 3,4,5,6,7,9,10,12,13,15,16,18,19,20,21,23,25,26,27,28,30,32,33,34,35,
%T A298006 37,39,40,41,42,44,46,47,49,50,52,53,54,55,57,58,59,61,63,64,66,67,69,
%U A298006 70,71,72,74,75,76,78,80,81,83,84,86,87,88,89,91,92,93
%N A298006 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, 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 A298006 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A297831.  See A297830 for a guide to related sequences.
%C A298006 Conjecture: 9/10 < a(n) - n*sqrt(2) < 3 for n >= 1.
%H A298006 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 A298006 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A298006 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 1;
%t A298006 j = 1; While[j < 80000, k = a[j] - j - 1;
%t A298006  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298006 u = Table[a[n], {n, 0, k}]; (* A297831 *)
%t A298006 v = Table[b[n], {n, 0, k}]; (* A298006 *)
%t A298006 Take[u, 50]
%t A298006 Take[v, 50]
%Y A298006 Cf. A297830, A297831.
%K A298006 nonn,easy
%O A298006 0,1
%A A298006 _Clark Kimberling_, Feb 09 2018