cp's OEIS Frontend

A296277 Solution of the complementary equation a(n) = a(n-1) + a(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 A296277 #4 Dec 13 2017 18:39:38
%S A296277 3,4,17,51,110,217,399,706,1215,2053,3424,5659,9293,15192,24773,40307,
%T A296277 65460,106187,172109,278802,451463,730865,1182978,1914545,3098279,
%U A296277 5013636,8112785,13127351,21241128,34369535,55611785,89982510,145595555,235579397,381176358
%N A296277 Solution of the complementary equation a(n) = a(n-1) + a(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 A296277 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 A296277 Clark Kimberling, <a href="/A296277/b296277.txt">Table of n, a(n) for n = 0..1000</a>
%H A296277 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 A296277 a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
%e A296277 a(2) = a(0) + a(1) + b(1)*b(2) = 17
%e A296277 Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, ...)
%t A296277 a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
%t A296277 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
%t A296277 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296277  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296277 Table[a[n], {n, 0, k}]; (* A296277 *)
%t A296277 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296277 Cf. A001622, A296245.
%K A296277 nonn,easy
%O A296277 0,1
%A A296277 _Clark Kimberling_, Dec 13 2017