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%I #28 Nov 06 2018 11:44:11
%S 1,2,10,13,16,19,22,25,29,34,38,43,47,52,56,61,65,70,74,79,82,86,91,
%T 94,97,101,106,109,113,118,121,124,128,133,136,140,145,148,151,155,
%U 160,163,167,172,175,178,182,187,190,194,199,202,205,209,214,217,221
%N Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
%C P. Majer proved that a(n)/n -> 4, that (a(n) - 4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964. - _Clark Kimberling_, Feb 10 2018
%H Peter J. C. Moses, <a href="/A296220/b296220.txt">Table of n, a(n) for n = 0..9999</a>
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%H Pietro Majer, <a href="https://mathoverflow.net/questions/290220/limit-associated-with-complementary-sequences">Limit associated with complementary sequences</a> Math Overflow.
%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
%e a(2) = a(0)*b(1) + a(1)*b(0) = 10.
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).
%t mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t a[0] = 1; a[1] = 2; b[0] = 3;
%t a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t u = Table[a[n], {n, 0, 500}]; (* A296220 *)
%t Table[b[n], {n, 0, 20}]
%Y Cf. A296000.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Dec 08 2017
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