A294424 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 5, 9, 14, 23, 38, 61, 99, 160, 260, 420, 680, 1100, 1780, 2880, 4660, 7540, 12201, 19741, 31942, 51683, 83625, 135308, 218933, 354241, 573174, 927415, 1500589, 2428004, 3928593, 6356597, 10285191, 16641788, 26926979, 43568767, 70495746, 114064513

Offset: 0

Author

Clark Kimberling, Nov 01 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) - 1 = 5
Complement: (b(n)) = (2, 4, 6, 7, 9, 11, 12, 13, 15,...)
		

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] - 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}]  (* A294424 *)
Table[b[n], {n, 0, 10}]