cp's OEIS Frontend

A296263 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296263 #7 Dec 12 2017 00:59:27
%S A296263 2,3,9,32,71,145,272,497,879,1508,2543,4233,6986,11459,18717,30482,
%T A296263 49541,80403,130364,211229,342099,553880,896579,1451109,2348390,
%U A296263 3800255,6149457,9950582,16100969,26052574,42154665,68208429,110364354,178574115,288939875
%N A296263 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296263 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 A296263 Clark Kimberling, <a href="/A296263/b296263.txt">Table of n, a(n) for n = 0..1000</a>
%H A296263 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 A296263 a(0) = 2, a(1) = 3, b(0) = 2, b(1) = 1, b(2) = 4
%e A296263 a(2) = a(0) + a(1) + b(0)*b(1) = 9
%e A296263 Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
%t A296263 a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
%t A296263 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
%t A296263 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296263 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296263 u = Table[a[n], {n, 0, k}];  (* A296263 *)
%t A296263 Table[b[n], {n, 0, 20}]      (* complement *)
%Y A296263 Cf. A001622, A296245.
%K A296263 nonn,easy
%O A296263 0,1
%A A296263 _Clark Kimberling_, Dec 11 2017