A294545 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 12, 24, 43, 75, 127, 212, 351, 576, 941, 1532, 2489, 4038, 6545, 10602, 17167, 27790, 44979, 72793, 117797, 190616, 308440, 499084, 807553, 1306667, 2114251, 3420950, 5535234, 8956218, 14491487, 23447741, 37939265, 61387044, 99326348

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..35.

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(1) + a(0) + b(1) -1 = 6.
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 13, 14, 15, 16, ...).

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] + 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}]  (* A294545 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A054061 A294563 A307211 * A305104 A118224 A227068

Adjacent sequences: A294542 A294543 A294544 * A294546 A294547 A294548

Keyword

nonn,easy

Author

Clark Kimberling, Nov 04 2017

Status

approved