A295613 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-1), where a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 3, 11, 27, 59, 116, 215, 383, 663, 1125, 1882, 3117, 5126, 8388, 13678, 22250, 36133, 58610, 94993, 153877, 249169, 403371, 652893, 1056646, 1709951, 2767040, 4477466, 7245014, 11723022, 18968613, 30692248, 49661511, 80354447, 130016685, 210371899

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Guide to related sequences:

A295613: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.

A295614: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.

A295615: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.

A295616: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.

A295617: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.

A295618: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

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, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, so that
b(3) = 7 (least "new number")
a(3) = 2*a(2) - a(0) + b(2) = 11
Complement: (b(n)) = (4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6;
a[n_] := a[n] = 2 a[n - 1] - a[n - 3] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 30}]   (* A295613 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045.

Sequence in context: A259428 A002981 A294637 * A232212 A232219 A265095

Adjacent sequences: A295610 A295611 A295612 * A295614 A295615 A295616

Keyword

nonn,easy

Author

Clark Kimberling, Nov 25 2017

Status

approved