cp's OEIS Frontend

A296276 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296276 #4 Dec 13 2017 18:39:31
%S A296276 2,4,21,55,118,229,419,738,1267,2137,3560,5879,9649,15768,25689,41763,
%T A296276 67794,109937,178171,288614,467337,756551,1224538,1981791,3207085,
%U A296276 5189688,8397643,13588261,21986896,35576213,57564231,93141634,150707125,243850091,394558622
%N A296276 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296276 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 A296276 Clark Kimberling, <a href="/A296276/b296276.txt">Table of n, a(n) for n = 0..1000</a>
%H A296276 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 A296276 a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5
%e A296276 a(2) = a(0) + a(1) + b(1)*b(2) = 21
%e A296276 Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)
%t A296276 a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; b[2] = 5;
%t A296276 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
%t A296276 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296276  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296276 Table[a[n], {n, 0, k}]; (* A296276 *)
%t A296276 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296276 Cf. A001622, A296245.
%K A296276 nonn,easy
%O A296276 0,1
%A A296276 _Clark Kimberling_, Dec 13 2017