A294561 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 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, 14, 30, 61, 111, 195, 332, 556, 920, 1511, 2469, 4023, 6539, 10612, 17204, 27872, 45135, 73069, 118269, 191406, 309746, 501226, 811049, 1312355, 2123487, 3435928, 5559506, 8995529, 14555133, 23550763, 38106000, 61656870, 99762980, 161419963, 261183059

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) + 2*b(1) + b(0) = 14
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294561 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A075718 A259710 A093794 * A075490 A031301 A335200

Adjacent sequences: A294558 A294559 A294560 * A294562 A294563 A294564

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved