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A297836 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*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 A297836 #4 Feb 06 2018 19:28:01
%S A297836 1,2,11,15,19,23,27,31,35,41,44,48,54,57,61,67,70,74,80,83,87,93,96,
%T A297836 100,106,109,113,119,122,126,130,134,140,143,149,152,156,162,165,169,
%U A297836 173,177,183,186,192,195,199,205,208,212,216,220,226,229,235,238,242
%N A297836 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*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 A297836 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. For a guide to related sequences, see A297830.
%C A297836 Conjectures:  a(n) - (5 + sqrt(13))*n/2 < 2 for n >= 1.
%H A297836 Clark Kimberling, <a href="/A297836/b297836.txt">Table of n, a(n) for n = 0..10000</a>
%e A297836 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 11.
%e A297836 Complement: (b(n)) = (3,4,5,6,7,8,9,10,12,13,14,16,17,18,20,...)
%t A297836 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A297836 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 3 n;
%t A297836 j = 1; While[j < 100, k = a[j] - j - 1;
%t A297836  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A297836 Table[a[n], {n, 0, k}]  (* A297836 *)
%Y A297836 Cf. A297826, A297830, A297837.
%K A297836 nonn,easy
%O A297836 0,2
%A A297836 _Clark Kimberling_, Feb 04 2018