A294398 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 7, 13, 20, 28, 38, 49, 61, 74, 88, 104, 121, 139, 158, 178, 199, 222, 246, 271, 297, 324, 352, 381, 412, 444, 477, 511, 546, 582, 619, 657, 696, 737, 779, 822, 866, 911, 957, 1004, 1052, 1101, 1151, 1203, 1256, 1310, 1365, 1421, 1478, 1536, 1595, 1655

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.

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) + 2 = 7
Complement: (b(n)) = (2, 4, 5, 6, 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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294398 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A022940.

Sequence in context: A330407 A014283 A341299 * A330707 A033551 A350296

Adjacent sequences: A294395 A294396 A294397 * A294399 A294400 A294401

Keyword

nonn,easy

Author

Clark Kimberling, Oct 30 2017

Status

approved