A294365 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 21, 41, 74, 129, 219, 367, 607, 997, 1629, 2653, 4311, 6995, 11339, 18369, 29745, 48154, 77941, 126139, 204126, 330313, 534489, 864854, 1399397, 2264307, 3663762, 5928129, 9591953, 15520146, 25112165, 40632379, 65744614, 106377065, 172121753

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) + 2 = 10;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 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] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294365 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293765.

Sequence in context: A027917 A038347 A338089 * A210980 A207380 A268348

Adjacent sequences: A294362 A294363 A294364 * A294366 A294367 A294368

Keyword

nonn,easy

Author

Clark Kimberling, Oct 29 2017

Status

approved