A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 22, 43, 78, 136, 231, 387, 641, 1053, 1721, 2803, 4555, 7391, 11981, 19409, 31429, 50879, 82352, 133278, 215679, 349008, 564740, 913803, 1478600, 2392462, 3871123, 6263648, 10134836, 16398551, 26533456, 42932078, 69465607, 112397760, 181863444

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

A294413: a(n) = a(n-1) + a(n-2) - b(n-1) + 6

A294414: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2)

A294415: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1

A294416: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n

A294417: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n

A294418: a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2)

A294419: a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2)

A294420: a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2)

A294421: a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2)

A294422: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1

A294423: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n

A294424: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 1

A294425: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2

A294426: a(n) = a(n-1) + 2*a(n-2) + b(n-1) + b(n-2) - 3

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

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) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(1) + b(0) = 6
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294414 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A022940.

Sequence in context: A006503 A248851 A023554 * A299336 A222629 A070880

Adjacent sequences: A294411 A294412 A294413 * A294415 A294416 A294417

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved