A293316 - OEIS

%I #12 Nov 04 2017 05:49:25

%S 1,3,7,15,28,50,87,147,245,404,662,1080,1757,2854,4629,7502,12151,

%T 19674,31847,51544,83415,134984,218425,353436,571889,925355,1497275,

%U 2422662,3919970,6342666,10262671,16605373,26868081,43473492,70341612,113815144,184156797

%N 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 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 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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

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

%e a(3) = a(2) + a(1) + b(1) + 1 = 13.

%e Complement: (b(n)) = (2,4,5,6,8,9,10,11,12,13,14,16,...).

%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 - 2] + 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}]  (* A293316 *)

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

%Y Cf. A001622 (golden ratio), A293076.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 28 2017