A294862 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 8, 13, 17, 24, 29, 37, 43, 53, 60, 71, 80, 92, 102, 115, 126, 140, 153, 168, 182, 198, 214, 231, 248, 266, 284, 303, 322, 343, 363, 385, 406, 429, 452, 476, 500, 525, 550, 576, 602, 629, 656, 685, 713, 743, 772, 803, 833, 866, 897, 931, 963, 998

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.

Links

Table of n, a(n) for n=0..54.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = a(0) + b(0) + 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 9, 10, 11, 12, 14, 15, 16, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + b[n - 2] + 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}]  (* A294862 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860, A294863.

Sequence in context: A054248 A038108 A356217 * A087327 A266627 A289753

Adjacent sequences: A294859 A294860 A294861 * A294863 A294864 A294865

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved