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A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*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 A297837 #4 Feb 06 2018 19:28:09
%S A297837 1,2,13,18,23,28,33,38,43,48,53,60,64,69,74,81,85,90,95,102,106,111,
%T A297837 116,123,127,132,137,144,148,153,158,165,169,174,179,186,190,195,200,
%U A297837 207,211,216,221,228,232,237,242,247,252,259,263,268,275,279,284,289
%N A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*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 A297837 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 A297837 Conjecture:  a(n) - (3 + sqrt(5))*n < 3 for n >= 1.
%H A297837 Clark Kimberling, <a href="/A297837/b297837.txt">Table of n, a(n) for n = 0..10000</a>
%e A297837 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 13.
%e A297837 Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...)
%t A297837 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A297837 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 4 n;
%t A297837 j = 1; While[j < 100, k = a[j] - j - 1;
%t A297837  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A297837 Table[a[n], {n, 0, k}]  (* A297836 *)
%Y A297837 Cf. A297826, A297830, A297836.
%K A297837 nonn,easy
%O A297837 0,2
%A A297837 _Clark Kimberling_, Feb 04 2018