cp's OEIS Frontend

A298001 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.

%I A298001 #4 Feb 08 2018 21:48:52
%S A298001 1,2,12,16,20,24,28,32,36,42,45,49,55,58,62,68,71,75,81,84,88,94,97,
%T A298001 101,107,110,114,120,123,127,131,135,141,144,150,153,157,163,166,170,
%U A298001 174,178,184,187,193,196,200,206,209,213,217,221,227,230,236,239,243
%N A298001 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 A298001 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 A298001 Conjecture:  a(n) - n*L < 4 for n >= 1, where L = (5 + sqrt(13))/2.
%H A298001 Clark Kimberling, <a href="/A298001/b298001.txt">Table of n, a(n) for n = 0..10000</a>
%e A298001 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 12.
%e A298001 Complement: (b(n)) = (3,4,5,6,8,9,10,11,14,15,17,18,19,21...)
%t A298001 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A298001 a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 3 n;
%t A298001 j = 1; While[j < 100, k = a[j] - j - 1;
%t A298001  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298001 Table[a[n], {n, 0, k}]  (* A298001 *)
%Y A298001 Cf. A297800, A297826, A297836, A297837.
%K A298001 nonn,easy
%O A298001 0,2
%A A298001 _Clark Kimberling_, Feb 08 2018