cp's OEIS Frontend

A296273 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296273 #6 Dec 31 2017 01:19:19
%S A296273 1,3,24,57,123,236,431,757,1298,2187,3641,6010,9861,16111,26244,42661,
%T A296273 69247,112288,181955,294705,477166,772446,1250262,2023410,3274428,
%U A296273 5298650,8573948,13873528,22448468,36323052,58772642,95096884,153870786,248969002,402841194
%N A296273 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296273 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 A296273 Clark Kimberling, <a href="/A296273/b296273.txt">Table of n, a(n) for n = 0..1000</a>
%H A296273 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 A296273 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5;
%e A296273 a(2) = a(0) + a(1) + b(1)*b(2) = 24;
%e A296273 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)
%t A296273 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296273 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
%t A296273 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296273 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296273 Table[a[n], {n, 0, k}];  (* A296273 *)
%t A296273 Table[b[n], {n, 0, 20}]  (* complement *)
%Y A296273 Cf. A001622, A296245.
%K A296273 nonn,easy
%O A296273 0,2
%A A296273 _Clark Kimberling_, Dec 12 2017