cp's OEIS Frontend

A298002 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*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 A298002 #4 Feb 08 2018 21:48:59
%S A298002 1,2,14,19,24,29,34,39,44,49,54,61,65,70,75,82,86,91,96,103,107,112,
%T A298002 117,124,128,133,138,145,149,154,159,166,170,175,180,187,191,196,201,
%U A298002 208,212,217,222,229,233,238,243,248,253,260,264,269,276,280,285,290
%N A298002 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*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 A298002 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 A298002 Conjecture:  a(n) - n*L < 4 for n >= 1, where L = 3 + sqrt(5).
%H A298002 Clark Kimberling, <a href="/A298002/b298002.txt">Table of n, a(n) for n = 0..10000</a>
%e A298002 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 14.
%e A298002 Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,13,15,17,18,20...)
%t A298002 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A298002 a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 4 n;
%t A298002 j = 1; While[j < 100, k = a[j] - j - 1;
%t A298002  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298002 Table[a[n], {n, 0, k}]  (* A298002 *)
%Y A298002 Cf. A297800, A297826, A297836, A297837.
%K A298002 nonn,easy
%O A298002 0,2
%A A298002 _Clark Kimberling_, Feb 08 2018