A022940 a(n) = a(n-1) + b(n-2) for n >= 3, a( ) increasing, given a(1) = 1, a(2) = 3; where b( ) is complement of a( ).

1, 3, 5, 9, 15, 22, 30, 40, 51, 63, 76, 90, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 354, 383, 414, 446, 479, 513, 548, 584, 621, 659, 698, 739, 781, 824, 868, 913, 959, 1006, 1054, 1103, 1153, 1205, 1258, 1312, 1367, 1423, 1480

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

From Clark Kimberling, Oct 30 2017: (Start)

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:

here: a(n) = a(n-1) + b(n-2) [with a different offset]

A294397: a(n) = a(n-1) + b(n-2) + 1;

A294398: a(n) = a(n-1) + b(n-2) + 2;

A294399: a(n) = a(n-1) + b(n-2) + 3;

A294400: a(n) = a(n-1) + b(n-2) + n;

A294401: a(n) = a(n-1) + b(n-2) + 2*n.

(End)

Examples

			a(1) = 1, a(2) = 3, b(1) = 2, b(2) = 4, so that a(3) = a(2) + a(1) + b(2) = 5.
Complement: {b(n)} = {2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, ...}
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A005228 and references therein.

Cf. A293076, A293765, A294381.

Programs

Mathematica

Fold[Append[#1, #1[[-1]] + Complement[Range[Max@#1 + 1], #1][[#2]]] &, {1, 3}, Range[50]] (* Ivan Neretin, Apr 04 2016 *)