cp's OEIS Frontend

A293316 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293316 #12 Nov 04 2017 05:49:25
%S A293316 1,3,7,15,28,50,87,147,245,404,662,1080,1757,2854,4629,7502,12151,
%T A293316 19674,31847,51544,83415,134984,218425,353436,571889,925355,1497275,
%U A293316 2422662,3919970,6342666,10262671,16605373,26868081,43473492,70341612,113815144,184156797
%N A293316 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293316 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 for a guide to related sequences.
%C A293316 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%e A293316 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293316 a(2) = a(1) + a(0) + b(0) + 1 = 7;
%e A293316 a(3) = a(2) + a(1) + b(1) + 1 = 13.
%e A293316 Complement: (b(n)) = (2,4,5,6,8,9,10,11,12,13,14,16,...).
%t A293316 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293316 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293316 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 1;
%t A293316 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293316 Table[a[n], {n, 0, 40}]  (* A293316 *)
%t A293316 Table[b[n], {n, 0, 10}]
%Y A293316 Cf. A001622 (golden ratio), A293076.
%K A293316 nonn,easy
%O A293316 0,2
%A A293316 _Clark Kimberling_, Oct 28 2017