A298001 - OEIS

%I #4 Feb 08 2018 21:48:52

%S 1,2,12,16,20,24,28,32,36,42,45,49,55,58,62,68,71,75,81,84,88,94,97,

%T 101,107,110,114,120,123,127,131,135,141,144,150,153,157,163,166,170,

%U 174,178,184,187,193,196,200,206,209,213,217,221,227,230,236,239,243

%N Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, 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 A298000 for a guide to related sequences.

%C Conjecture:  a(n) - n*L < 4 for n >= 1, where L = (5 + sqrt(13))/2.

%H Clark Kimberling, <a href="/A298001/b298001.txt">Table of n, a(n) for n = 0..10000</a>

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 12.

%e Complement: (b(n)) = (3,4,5,6,8,9,10,11,14,15,17,18,19,21...)

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

%t a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 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}]  (* A298001 *)

%Y Cf. A297800, A297826, A297836, A297837.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 08 2018