cp's OEIS Frontend

A293767 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293767 #7 Nov 02 2017 09:20:37
%S A293767 1,3,7,14,26,47,81,137,228,376,616,1006,1637,2659,4313,6990,11322,
%T A293767 18332,29675,48029,77727,125780,203533,329340,532901,862270,1395201,
%U A293767 2257502,3652735,5910270,9563039,15473344,25036419,40509800,65546257,106056096
%N A293767 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293767 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
%C A293767 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences.
%H A293767 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.pdf">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A293767 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293767 a(2) = a(1) + a(0) + b(1) - 1 = 7;
%e A293767 Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)
%t A293767 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293767 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293767 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - 1;
%t A293767 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293767 Table[a[n], {n, 0, 40}]  (* A293767 *)
%t A293767 Table[b[n], {n, 0, 10}]
%Y A293767 Cf. A001622 (golden ratio), A293765.
%K A293767 nonn,easy
%O A293767 0,2
%A A293767 _Clark Kimberling_, Oct 29 2017