A293767 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 7, 14, 26, 47, 81, 137, 228, 376, 616, 1006, 1637, 2659, 4313, 6990, 11322, 18332, 29675, 48029, 77727, 125780, 203533, 329340, 532901, 862270, 1395201, 2257502, 3652735, 5910270, 9563039, 15473344, 25036419, 40509800, 65546257, 106056096

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.

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) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) - 1 = 7;
Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)

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] - 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}]  (* A293767 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293765.

Sequence in context: A001924 A079921 A369115 * A014168 A132109 A317779

Adjacent sequences: A293764 A293765 A293766 * A293768 A293769 A293770

Keyword

nonn,easy

Author

Clark Kimberling, Oct 29 2017

Status

approved