A081689 -id:A081689 - cp's OEIS frontend

A081688 0 followed by A030124 - 1.

0, 1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 0

N. J. A. Sloane, Apr 02 2003

nonn

From P-positions in a certain game.

The rule "monotonically increasing sequence where the size of each run of consecutive integers is given by the sequence itself" produces this sequence without the initial 0. - Eric Angelini, Aug 19 2008

A. S. Fraenkel, Home Page
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A030124, A005228, A081689.

Let a(n) = this sequence, b(n) = A081689. Then a(n) = mex{ a(i), b(i) : 0 <= i < n}, b(n) = b(n-1) + a(n) + 1. Apart from initial zero, complement of A081689.

A294397 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 44, 55, 68, 82, 97, 113, 130, 149, 169, 190, 212, 235, 259, 284, 311, 339, 368, 398, 429, 461, 494, 528, 564, 601, 639, 678, 718, 759, 801, 844, 888, 934, 981, 1029, 1078, 1128, 1179, 1231, 1284, 1338, 1393, 1450, 1508, 1567, 1627

Offset: 0

Clark Kimberling, Oct 30 2017

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

Apart from the first two entries this is the same as A081689. - R. J. Mathar, Oct 31 2017

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + b(0) + 1 = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, ...)

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

Cf. A081689, A293076, A293765, A022940.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294397 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A081688 0 followed by A030124 - 1.

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