A294543 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 18, 35, 62, 107, 181, 301, 496, 812, 1324, 2153, 3495, 5667, 9183, 14872, 24078, 38974, 63077, 102077, 165181, 267286, 432496, 699812, 1132339, 1832183, 2964555, 4796772, 7761362, 12558170, 20319570, 32877779, 53197389, 86075209, 139272640

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) + 2 = 9.
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...).

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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294543 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A103256 A028881 A294535 * A295956 A296843 A200085

Adjacent sequences: A294540 A294541 A294542 * A294544 A294545 A294546

Keyword

nonn,easy

Author

Clark Kimberling, Nov 04 2017

Status

approved