cp's OEIS Frontend

A296268 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296268 #6 Dec 12 2017 16:37:42
%S A296268 1,4,15,37,87,172,322,574,995,1689,2827,4684,7719,12641,20648,33612,
%T A296268 54620,88631,143691,232805,377024,610404,988052,1599131,2587911,
%U A296268 4187825,6776576,10965300,17742836,28709159,46453083,75163397,121617704,196782431,318401539
%N A296268 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296268 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 A296268 Clark Kimberling, <a href="/A296268/b296268.txt">Table of n, a(n) for n = 0..1000</a>
%H A296268 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 A296268 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5;
%e A296268 a(2) = a(0) + a(1) + b(0)*b(2) = 15;
%e A296268 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, ...)
%t A296268 a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
%t A296268 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n];
%t A296268 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296268 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296268 Table[a[n], {n, 0, k}];  (* A296268 *)
%t A296268 Table[b[n], {n, 0, 20}]  (* complement *)
%Y A296268 Cf. A001622, A296245.
%K A296268 nonn,easy
%O A296268 0,2
%A A296268 _Clark Kimberling_, Dec 12 2017