A294559 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3793, 6166, 10008, 16226, 26289, 42573, 68923, 111560, 180550, 292180, 472803, 765059, 1237941, 2003083, 3241112, 5244286, 8485492, 13729875, 22215467, 35945445, 58161018, 94106572, 152267702, 246374389

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) + 2*b(0)  = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, ...)

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] + 2 b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294559 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A031090 A358296 A294556 * A041017 A033837 A041575

Adjacent sequences: A294556 A294557 A294558 * A294560 A294561 A294562

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved