A293351 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n -1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 7, 16, 31, 57, 101, 173, 291, 483, 795, 1301, 2121, 3449, 5600, 9081, 14715, 23832, 38585, 62457, 101084, 163585, 264715, 428348, 693113, 1121513, 1814680, 2936249, 4750988, 7687298, 12438349, 20125712, 32564128, 52689909, 85254108, 137944090

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

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] + 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}]  (* A293351 *)
Table[b[n], {n, 0, 10}]