A294366 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 12, 26, 52, 95, 167, 285, 478, 792, 1303, 2131, 3473, 5646, 9164, 14858, 24073, 38985, 63115, 102160, 165338, 267564, 432971, 700608, 1133655, 1834342, 2968079, 4802506, 7770673, 12573270, 20344037, 32917404, 53261541, 86179048, 139440695, 225619852

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) + 4 = 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] + 2n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294366 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293765.

Sequence in context: A199242 A326725 A169678 * A110859 A190904 A345960

Adjacent sequences: A294363 A294364 A294365 * A294367 A294368 A294369

Keyword

nonn,easy

Author

Clark Kimberling, Oct 29 2017

Status

approved