A294562 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) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 5, 10, 17, 29, 48, 80, 130, 212, 344, 558, 904, 1465, 2371, 3838, 6211, 10051, 16264, 26317, 42583, 68902, 111487, 180391, 291881, 472274, 764157, 1236433, 2000592, 3237027, 5237621, 8474650, 13712273, 22186925, 35899200, 58086127

Offset

0,2

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).

Links

Table of n, a(n) for n=0..35.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2)  = a(1) + a(0) + b(1) - b(0) + 1 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, ...)

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 - 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}]  (* A294562 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A178137 A071602 A046485 * A109377 A109472 A172167

Adjacent sequences: A294559 A294560 A294561 * A294563 A294564 A294565

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved