cp's OEIS Frontend

A297831 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, 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.

%I A297831 #4 Feb 06 2018 19:27:27
%S A297831 1,2,8,11,14,17,22,24,29,31,36,38,43,45,48,51,56,60,62,65,68,73,77,79,
%T A297831 82,85,90,94,96,99,102,107,111,113,118,120,125,127,130,133,138,140,
%U A297831 143,148,152,154,159,161,166,168,171,174,179,181,184,189,193,195
%N A297831 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, 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 A297831 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 A297831 Conjecture:  a(n) - (2 +sqrt(2))*n < 5/2 for n >= 1.
%H A297831 Clark Kimberling, <a href="/A297831/b297831.txt">Table of n, a(n) for n = 0..10000</a>
%e A297831 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 8.
%e A297831 Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...)
%t A297831 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A297831 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 1;
%t A297831 j = 1; While[j < 100, k = a[j] - j - 1;
%t A297831  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A297831 Table[a[n], {n, 0, k}]  (* A297831 *)
%Y A297831 Cf. A297826, A297830.
%K A297831 nonn,easy
%O A297831 0,2
%A A297831 _Clark Kimberling_, Feb 04 2018