cp's OEIS Frontend

A296262 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296262 #7 Dec 12 2017 00:59:17
%S A296262 1,4,11,30,71,143,270,485,845,1450,2451,4083,6744,11067,18083,29456,
%T A296262 47881,77717,126018,204197,330721,535470,866791,1402911,2270404,
%U A296262 3674071,5945287,9620257,15566536,25187849,40755507,65944546,106701313,172647191,279349910
%N A296262 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296262 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 A296262 Clark Kimberling, <a href="/A296262/b296262.txt">Table of n, a(n) for n = 0..999</a>
%H A296262 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 A296262 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3,
%e A296262 a(2) = a(0) + a(1) + b(0)*b(1) = 11
%e A296262 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
%t A296262 a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
%t A296262 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
%t A296262 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296262 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296262 u = Table[a[n], {n, 0, k}];  (* A296262 *)
%t A296262 Table[b[n], {n, 0, 20}]  (* complement *)
%Y A296262 Cf. A001622, A296245.
%K A296262 nonn,easy
%O A296262 0,2
%A A296262 _Clark Kimberling_, Dec 11 2017