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%I #16 May 01 2018 03:01:23
%S 1,2,13,16,19,22,25,28,31,34,38,43,47,52,56,61,65,70,74,79,83,88,92,
%T 97,101,106,109,113,118,121,124,128,133,136,140,145,148,151,155,160,
%U 163,167,172,175,178,182,187,190,194,199,202,205,209,214,217,221,226
%N Solution a( ) of the complementary equation a(n) = a(0)*b(n) + a(1)*b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, 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.
%H Clark Kimberling, <a href="/A298295/b298295.txt">Table of n, a(n) for n = 0..1000</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.
%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 13.
%e Complement: (3,4,5,6,7,8,9,10,11,12,14,15,17,...)
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[0]*b[n] + a[1]*b[n - 1]
%t Table[{a[n],
%t b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 1010}];
%t Table[a[n], {n, 0, 150}] (* A298295 *)
%t Table[b[n], {n, 0, 150}] (* A298296 *)
%t (* _Peter J. C. Moses_, Jan 16 2018 *)
%Y Cf. A298296, A297830, A298000.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Feb 09 2018
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