A294413 Solution of the complementary equation a(n) = a(n-1) + a(n-2) - b(n-1) + 6, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 6, 10, 15, 23, 35, 53, 82, 128, 202, 320, 511, 819, 1317, 2122, 3424, 5530, 8936, 14447, 23363, 37789, 61130, 98896, 160002, 258873, 418849, 677695, 1096516, 1774181, 2870666, 4644815, 7515448, 12160229, 19675642, 31835835

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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) + 6
Complement: (b(n)) = (2, 4, 6, 7, 9, 11, 12, 13, 14, 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] + 6;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294413 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A294414.

Sequence in context: A342211 A262927 A063542 * A122554 A111734 A372111

Adjacent sequences: A294410 A294411 A294412 * A294414 A294415 A294416

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2017

Status

approved