A294565 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 12, 25, 44, 77, 130, 217, 360, 590, 964, 1569, 2549, 4135, 6702, 10856, 17578, 28455, 46055, 74533, 120614, 195173, 315814, 511015, 826858, 1337903, 2164792, 3502727, 5667552, 9170313, 14837900, 24008249, 38846186, 62854473, 101700698, 164555211

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..36.

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) + 2*b(1) - b(0) - 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 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;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 2 b[n - 1] - b[n - 2] - 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}]  (* A294565 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A034882 A175943 A228816 * A210068 A210633 A304710

Adjacent sequences: A294562 A294563 A294564 * A294566 A294567 A294568

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved