A294560 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 17, 37, 76, 139, 245, 418, 701, 1161, 1908, 3119, 5081, 8258, 13401, 21727, 35202, 57007, 92291, 149384, 241765, 391243, 633106, 1024451, 1657663, 2682224, 4340001, 7022343, 11362466, 18384935, 29747531, 48132600, 77880269, 126013011, 203893428

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..34.

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(0) + a(1) + 2*b(0) + 2*b(1) = 17
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 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] + 2 b[n - 1] + 2 b[n - 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}]  (* A294560 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A041965 A095075 A307161 * A258977 A069042 A121923

Adjacent sequences: A294557 A294558 A294559 * A294561 A294562 A294563

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved