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.

1, 3, 7, 15, 28, 50, 87, 147, 245, 404, 662, 1080, 1757, 2854, 4629, 7502, 12151, 19674, 31847, 51544, 83415, 134984, 218425, 353436, 571889, 925355, 1497275, 2422662, 3919970, 6342666, 10262671, 16605373, 26868081, 43473492, 70341612, 113815144, 184156797

Offset: 0

Author

Clark Kimberling, Oct 28 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 for a guide to related sequences.

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

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(0) + 1 = 7;
a(3) = a(2) + a(1) + b(1) + 1 = 13.
Complement: (b(n)) = (2,4,5,6,8,9,10,11,12,13,14,16,...).
		

Crossrefs

Cf. A001622 (golden ratio), A293076.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293316 *)
Table[b[n], {n, 0, 10}]