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.

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

0,2

Comments

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

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) = 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, ...)

Mathematica

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}]

Crossrefs

Cf. A081689, A293076, A293765, A022940.

Sequence in context: A023601 A173143 A109413 * A003022 A025722 A022775

Adjacent sequences: A294394 A294395 A294396 * A294398 A294399 A294400

Keyword

nonn,easy

Author

Clark Kimberling, Oct 30 2017

Status

approved