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A297830 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n, 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 A297830 #6 Sep 30 2020 05:03:14
%S A297830 1,2,9,12,15,18,21,26,28,33,35,40,42,47,49,54,56,59,62,67,71,73,76,79,
%T A297830 84,88,90,93,96,101,105,107,110,113,118,122,124,127,130,135,139,141,
%U A297830 146,148,153,155,158,161,166,168,171,176,180,182,187,189,194,196
%N A297830 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n, 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 A297830 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Conjecture: a(n) - (2 +sqrt(2))*n < 3 for n >= 1.
%C A297830 Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
%C A297830 ***
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n           (a(n)) = A297826; (b(n)) = A297997
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n         (a(n)) = A297830; (b(n)) = A298003
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n         (a(n)) = A297836; (b(n)) = A298004
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n         (a(n)) = A297837; (b(n)) = A298005
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1     (a(n)) = A297831; (b(n)) = A298006
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2     (a(n)) = A297832; (b(n)) = A298007
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3     (a(n)) = A297833; (b(n)) = A298108
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4     (a(n)) = A297834; (b(n)) = A298109
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1     (a(n)) = A297835;
%C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2)  (a(n)) = A297998;
%C A297830 ***
%C A297830 For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800.
%H A297830 Clark Kimberling, <a href="/A297830/b297830.txt">Table of n, a(n) for n = 0..10000</a>
%e A297830 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9.
%e A297830 Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...)
%t A297830 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A297830 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n;
%t A297830 j = 1; While[j < 100, k = a[j] - j - 1;
%t A297830 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t A297830 Table[a[n], {n, 0, k}]  (* A297830 *)
%Y A297830 Cf. A297826, A297836, A297837.
%K A297830 nonn,easy
%O A297830 0,2
%A A297830 _Clark Kimberling_, Feb 04 2018