A295070 Solution of the complementary equation a(n) = a(n-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, 8, 11, 19, 24, 35, 43, 57, 68, 84, 97, 115, 130, 150, 168, 191, 211, 236, 259, 287, 312, 342, 369, 401, 430, 464, 495, 531, 565, 604, 640, 681, 719, 762, 802, 848, 891, 939, 984, 1034, 1081, 1133, 1182, 1236, 1287, 1343, 1396, 1454, 1510, 1571, 1629

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> 1.

Links

Table of n, a(n) for n=0..51.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Example

a(0) = 1, a(1) = 2, b(0) = 3
a(2) = a(0) + b(1) + b(0) = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, ... )

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 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, 18}]  (* A295070 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A143189 A056550 A074263 * A009420 A189328 A090746

Adjacent sequences: A295067 A295068 A295069 * A295071 A295072 A295073

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved