cp's OEIS Frontend

A296270 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*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 A296270 #6 Dec 12 2017 16:37:56
%S A296270 2,4,11,33,79,160,302,542,952,1624,2744,4563,7531,12349,20168,32840,
%T A296270 53368,86607,140415,227505,368448,596528,965600,1562803,2529131,
%U A296270 4092717,6622688,10716304,17339952,28057310,45398382,73456916,118856593,192314877,311172913
%N A296270 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*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 A296270 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 A296270 Clark Kimberling, <a href="/A296270/b296270.txt">Table of n, a(n) for n = 0..1000</a>
%H A296270 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 A296270 a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5;
%e A296270 a(2) = a(0) + a(1) + b(0)*b(2) = 11;
%e A296270 Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
%t A296270 a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; b[2] = 5;
%t A296270 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n];
%t A296270 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296270 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296270 Table[a[n], {n, 0, k}];  (* A296270 *)
%t A296270 Table[b[n], {n, 0, 20}]  (* complement *)
%Y A296270 Cf. A001622, A296245.
%K A296270 nonn,easy
%O A296270 0,1
%A A296270 _Clark Kimberling_, Dec 12 2017