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A296846 Solution of the complementary equation a(n) = a(n-1) + a(n-2) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296846 #8 Feb 11 2018 04:26:49
%S A296846 3,5,7,10,13,17,22,30,41,59,86,130,200,312,493,785,1257,2019,3252,
%T A296846 5246,8472,13691,22135,35797,57901,93666,151534,245166,396665,641795,
%U A296846 1038423,1680180,2718564,4398704,7117226,11515887,18633069,30148911,48781934,78930798
%N A296846 Solution of the complementary equation a(n) = a(n-1) + a(n-2) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296846 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.
%H A296846 Clark Kimberling, <a href="/A296846/b296846.txt">Table of n, a(n) for n = 0..999</a>
%H A296846 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 A296846 a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4
%e A296846 a(2) = a(0) + a(1) - b(0) = 7
%e A296846 Complement: (b(n)) = (1, 2, 4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, ...)
%t A296846 a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
%t A296846 a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
%t A296846 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296846 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296846 u = Table[a[n], {n, 0, k}];  (* A296846 *)
%t A296846 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296846 Cf. A001622, A296245, A296847.
%K A296846 nonn,easy
%O A296846 0,1
%A A296846 _Clark Kimberling_, Jan 12 2018