cp's OEIS Frontend

A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296265 #6 Dec 12 2017 16:37:21
%S A296265 3,4,9,23,62,127,245,452,807,1391,2354,3927,6491,10658,17421,28385,
%T A296265 46148,74913,121481,196856,318865,516321,835836,1352859,2189451,
%U A296265 3543122,5733443,9277495,15011930,24290481,39303533,63595204,102899997,166496533,269397936
%N A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296265 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 A296265 Clark Kimberling, <a href="/A296265/b296265.txt">Table of n, a(n) for n = 0..999</a>
%H A296265 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 A296265 a(0) = 3, a(1) = 4, b(0) = 2, b(1) = 1, b(2) = 2;
%e A296265 a(2) = a(0) + a(1) + b(0)*b(1) = 9;
%e A296265 Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...)
%t A296265 a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2;
%t A296265 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
%t A296265 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296265 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296265 Table[a[n], {n, 0, k}];  (* A296265 *)
%t A296265 Table[b[n], {n, 0, 20}]  (* complement *)
%Y A296265 Cf. A001622, A296245.
%K A296265 nonn,easy
%O A296265 0,1
%A A296265 _Clark Kimberling_, Dec 12 2017