%I #4 Feb 06 2018 19:28:09

%S 1,2,13,18,23,28,33,38,43,48,53,60,64,69,74,81,85,90,95,102,106,111,

%T 116,123,127,132,137,144,148,153,158,165,169,174,179,186,190,195,200,

%U 207,211,216,221,228,232,237,242,247,252,259,263,268,275,279,284,289

%N 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 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 Conjecture: a(n) - (3 + sqrt(5))*n < 3 for n >= 1.

%H Clark Kimberling, <a href="/A297837/b297837.txt">Table of n, a(n) for n = 0..10000</a>

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 13.

%e Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...)

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;

%t a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 4 n;

%t j = 1; While[j < 100, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k

%t Table[a[n], {n, 0, k}] (* A297836 *)

%Y Cf. A297826, A297830, A297836.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 04 2018