A294549 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 21, 42, 76, 133, 226, 379, 627, 1030, 1683, 2741, 4454, 7227, 11715, 18978, 30731, 49750, 80524, 130319, 210890, 341258, 552199, 893510, 1445764, 2339331, 3785154, 6124546, 9909763, 16034374, 25944204, 41978647, 67922922, 109901642, 177824639

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) + 1 = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...).

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] + 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}]  (* A294549 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A156818 A037422 A136735 * A294550 A296555 A231376

Adjacent sequences: A294546 A294547 A294548 * A294550 A294551 A294552

Keyword

nonn,easy

Author

Clark Kimberling, Nov 04 2017

Status

approved