A297836 - OEIS

%I #4 Feb 06 2018 19:28:01

%S 1,2,11,15,19,23,27,31,35,41,44,48,54,57,61,67,70,74,80,83,87,93,96,

%T 100,106,109,113,119,122,126,130,134,140,143,149,152,156,162,165,169,

%U 173,177,183,186,192,195,199,205,208,212,216,220,226,229,235,238,242

%N 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 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 Conjectures:  a(n) - (5 + sqrt(13))*n/2 < 2 for n >= 1.

%H Clark Kimberling, <a href="/A297836/b297836.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) = 11.

%e Complement: (b(n)) = (3,4,5,6,7,8,9,10,12,13,14,16,17,18,20,...)

%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] + 3 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, A297837.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 04 2018