A294532 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 6, 12, 23, 42, 73, 124, 207, 342, 562, 918, 1495, 2429, 3941, 6388, 10348, 16756, 27125, 43903, 71052, 114980, 186058, 301065, 487151, 788245, 1275426, 2063702, 3339160, 5402895, 8742089, 14145019, 22887144, 37032200, 59919382, 96951621, 156871043

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3:

a(n) = a(n-1) + a(n-2) + b(n-2) A294532

a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533

a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534

a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535

a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536

a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537

a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538

a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539

a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540

a(n) = a(n-1) + a(n-2) + b(n-1) A294541

a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542

a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543

a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544

a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545

a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546

a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547

a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548

a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557

a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558

a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294559

a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294560

a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294561

a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562

a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563

a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1 A294564

a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 3 A294565

Conjecture: for every sequence listed here, 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(0) + a(1) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ...)

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 - 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}]  (* A294532 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A293076, A294413.

Sequence in context: A062476 A192703 A192969 * A323950 A291014 A257479

Adjacent sequences: A294529 A294530 A294531 * A294533 A294534 A294535

Keyword

nonn,easy

Author

Clark Kimberling, Nov 03 2017

Status

approved