A297835 - OEIS

%I #4 Feb 06 2018 19:27:55

%S 1,2,10,13,16,19,22,25,30,32,37,39,44,46,51,53,58,60,65,67,70,73,78,

%T 82,84,87,90,95,99,101,104,107,112,116,118,121,124,129,133,135,138,

%U 141,146,150,152,155,158,163,167,169,174,176,181,183,186,189,194,196

%N Solution 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 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences.

%C Conjecture:  a(n) - (2 +sqrt(2))*n < 7 for n >= 1.

%H Clark Kimberling, <a href="/A297835/b297835.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) = 10.

%e Complement: (b(n)) = (3,4,6,7,8,9,11,12,14,15,17,18,20,...)

%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] + 2 n + 1;

%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}]  (* A297835 *)

%Y Cf. A297826, A297830.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 04 2018