A293767 - OEIS

%I #7 Nov 02 2017 09:20:37

%S 1,3,7,14,26,47,81,137,228,376,616,1006,1637,2659,4313,6990,11322,

%T 18332,29675,48029,77727,125780,203533,329340,532901,862270,1395201,

%U 2257502,3652735,5910270,9563039,15473344,25036419,40509800,65546257,106056096

%N 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 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 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences.

%H 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 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

%e a(2) = a(1) + a(0) + b(1) - 1 = 7;

%e Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - 1;

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 40}]  (* A293767 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A001622 (golden ratio), A293765.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 29 2017