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A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296283 #4 Dec 13 2017 18:40:23
%S A296283 3,4,17,81,308,725,1537,2982,5509,9811,17036,29031,48797,81188,134305,
%T A296283 220965,362110,591055,962405,1564086,2538635,4116521,6670756,10804827,
%U A296283 17495239,28321990,45841589,74190549,120061898,194285183,314382985,508707438,823133263
%N A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296283 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 A296283 Clark Kimberling, <a href="/A296283/b296283.txt">Table of n, a(n) for n = 0..1000</a>
%H A296283 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 A296283 a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
%e A296283 a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 17
%e A296283 Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, ...)
%t A296283 a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
%t A296283 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2] b[n - 1] b[n];
%t A296283 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296283  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296283 Table[a[n], {n, 0, k}]; (* A296283 *)
%t A296283 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296283 Cf. A001622, A296245.
%K A296283 nonn,easy
%O A296283 0,1
%A A296283 _Clark Kimberling_, Dec 13 2017