A295363 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 12, 35, 77, 154, 287, 513, 890, 1513, 2546, 4241, 6997, 11478, 18747, 30531, 49620, 80531, 130571, 211564, 342641, 554757, 897998, 1453405, 2352105, 3806266, 6159183, 9966319, 16126432, 26093743, 42221231

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.

Links

Table of n, a(n) for n=0..30.

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

Formula

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
b(3) = 5 (least "new number")
a(2) = a(1) + a(0) + b(0)*b(1) = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)

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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 32; u = Table[a[n], {n, 0, z}]   (* A295363 *)
v = Table[b[n], {n, 0, 10}]  (* complement *)

Crossrefs

Cf. A001622, A295357.

Sequence in context: A326660 A225075 A293267 * A097339 A260006 A303862

Adjacent sequences: A295360 A295361 A295362 * A295364 A295365 A295366

Keyword

nonn,easy

Author

Clark Kimberling, Nov 21 2017

Status

approved