A294536 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 5, 10, 20, 36, 63, 107, 180, 298, 490, 801, 1305, 2121, 3442, 5580, 9040, 14640, 23701, 38363, 62087, 100474, 162586, 263086, 425699, 688813, 1114541, 1803384, 2917956, 4721372, 7639361, 12360767, 20000164, 32360968, 52361170, 84722177, 137083387

Offset: 0

Author

Clark Kimberling, Nov 03 2017

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622)..

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2)  = a(1) + a(0) + b(0) - 1 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, ...)
		

Links

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

Crossrefs

Cf. A001622, A294532.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 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}]  (* A294536 *)
Table[b[n], {n, 0, 10}]