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A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*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 A298000 #4 Feb 06 2018 19:28:22
%S A298000 1,2,10,13,16,19,22,27,29,34,36,41,43,48,50,55,57,60,63,68,72,74,77,
%T A298000 80,85,89,91,94,97,102,106,108,111,114,119,123,125,128,131,136,140,
%U A298000 142,147,149,154,156,159,162,167,169,172,177,181,183,188,190,195,197
%N A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*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 A298000 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.
%C A298000 Conjectures:  a(n) - (2 +sqrt(2))*n < 4 for n >= 1.  Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
%C A298000 ***
%C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + n     (a(n)) = A297999; (b(n)) = A298110
%C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n   (a(n)) = A298000; (b(n)) = A298111
%C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n   (a(n)) = A298001; (b(n)) = A298112
%C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n   (a(n)) = A298002; (b(n)) = A298113
%H A298000 Clark Kimberling, <a href="/A298000/b298000.txt">Table of n, a(n) for n = 0..10000</a>
%e A298000 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10.
%e A298000 Complement: (b(n)) = (3,4,5,6,8,9,11,12,14,15,17,18,20,...)
%t A298000 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A298000 a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 n;
%t A298000 j = 1; While[j < 100, k = a[j] - j - 1;
%t A298000  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A298000 Table[a[n], {n, 0, k}]  (* A298000 *)
%Y A298000 Cf. A297826, A297836, A297837.
%K A298000 nonn,easy
%O A298000 0,2
%A A298000 _Clark Kimberling_, Feb 04 2018