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%I #4 Feb 06 2018 19:27:42
%S 1,2,6,9,14,16,21,23,26,29,34,38,40,43,46,51,55,57,62,64,69,71,74,77,
%T 82,84,87,92,96,98,103,105,110,112,115,118,123,125,128,133,137,139,
%U 142,145,150,154,156,159,162,167,171,173,178,180,185,187,190,193
%N Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3, 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. See A297830 for a guide to related sequences.
%C Conjecture: -2 < a(n) - (2 +sqrt(2))*n <= 1 for n >= 1.
%H Clark Kimberling, <a href="/A297833/b297833.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) = 6.
%e Complement: (b(n)) = (3,4,5,7,8,10,12,13,15,17,18,19,...)
%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] + 2 n - 3;
%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}] (* A297833 *)
%Y Cf. A297826, A297830.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Feb 04 2018
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