A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 6, 13, 24, 44, 76, 129, 215, 355, 582, 951, 1548, 2515, 4080, 6613, 10712, 17345, 28078, 45445, 73546, 119016, 192588, 311631, 504247, 815907, 1320184, 2136122, 3456338, 5592493, 9048865, 14641393, 23690294, 38331724, 62022056, 100353819, 162375915

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, which for each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

A293076: a(n) = a(n-1) + a(n-2) + b(n-2)

A293316: a(n) = a(n-1) + a(n-2) + b(n-2) + 1

A293057: a(n) = a(n-1) + a(n-2) + b(n-2) + 2

A293058: a(n) = a(n-1) + a(n-2) + b(n-2) + 3

A293317: a(n) = a(n-1) + a(n-2) + b(n-2) - 1

A293349: a(n) = a(n-1) + a(n-2) + b(n-2) + n

A293350: a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n

A293351: a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1

A293357: a(n) = a(n-1) + a(n-2) + b(n-2) + 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(0) = 3 + 1 + 2 = 6;
a(3) = a(2) + a(1) + b(1) = 6 + 3 + 4 = 13.
Complement: (b(n)) = (2,4,5,7,8,9,10,11,12,14,...)
		

Links

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

Crossrefs

Cf. A001622 (golden ratio), A293358.

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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293076 *)
Table[b[n], {n, 0, 10}]

Extensions

Comments corrected by Georg Fischer, Sep 23 2020