A294423 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 15, 28, 49, 85, 142, 236, 388, 635, 1035, 1684, 2733, 4432, 7181, 11630, 18829, 30478, 49327, 79826, 129175, 209024, 338223, 547273, 885522, 1432822, 2318372, 3751223, 6069625, 9820879, 15890536, 25711448, 41602018, 67313501, 108915555, 176229093

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) - b(0)  + 2 = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 11, 13, ...)

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] - b[n - 2] + 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}]  (* A294423 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A294414.

Sequence in context: A036419 A054107 A031125 * A294426 A097589 A015631

Adjacent sequences: A294420 A294421 A294422 * A294424 A294425 A294426

Keyword

nonn,easy

Author

Clark Kimberling, Nov 01 2017

Status

approved