A293358 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 16, 30, 53, 92, 155, 258, 425, 696, 1135, 1846, 2998, 4862, 7879, 12761, 20661, 33444, 54128, 87596, 141749, 229371, 371147, 600546, 971722, 1572299, 2544053, 4116385, 6660472, 10776892, 17437400, 28214329, 45651767, 73866135, 119517942

Offset: 0

Author

Clark Kimberling, Oct 29 2017

Keywords

• nonn
• easy

Comments

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:

A293358: a(n) = a(n-1) + a(n-2) + b(n-1)

A293406: a(n) = a(n-1) + a(n-2) + b(n-1) + 1

A293765: a(n) = a(n-1) + a(n-2) + b(n-1) + 2

A293766: a(n) = a(n-1) + a(n-2) + b(n-1) + 3

A293767: a(n) = a(n-1) + a(n-2) + b(n-1) - 1

A294365: a(n) = a(n-1) + a(n-2) + b(n-1) + n

A294366: a(n) = a(n-1) + a(n-2) + b(n-1) + 2n

A294367: a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1

A294368: a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1

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(1) = 8;
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...)
		

Links

• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

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 - 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}]  (* A293358 *)
Table[b[n], {n, 0, 10}]