A294416 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 12, 27, 54, 99, 174, 297, 498, 825, 1357, 2220, 3618, 5882, 9547, 15479, 25079, 40614, 65752, 106428, 172245, 278741, 451057, 729872, 1181007, 1910961, 3092053, 5003102, 8095246, 13098442, 21193785, 34292327, 55486215, 89778648, 145264972, 235043732

Offset: 0

Author

Clark Kimberling, Oct 31 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 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(1) + b(0) + 2 = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14,...)
		

Links

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

Crossrefs

Cf. A293076, A293765, A294414.

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 - 2] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294416 *)
Table[b[n], {n, 0, 10}]