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A297834 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, 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. 3

%I

%S 1,2,5,8,12,17,19,22,27,29,32,35,40,44,46,51,53,56,59,64,68,70,75,77,

%T 82,84,87,90,95,97,100,105,109,111,114,117,122,126,128,133,135,140,

%U 142,145,148,153,155,158,163,167,169,172,175,180,184,186,189,192

%N Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, 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: -3 < a(n) - (2 +sqrt(2))*n <= 1 for n >= 1.

%H Clark Kimberling, <a href="/A297834/b297834.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) = 5.

%e Complement: (b(n)) = (3,4,6,7,9,10,11,13,14,15,16,18,20,...)

%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 - 4;

%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}] (* A297834 *)

%Y Cf. A297826, A297830.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 04 2018

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Last modified September 18 12:35 EDT 2021. Contains 347527 sequences. (Running on oeis4.)