A294867 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) -1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 14, 28, 49, 78, 116, 164, 223, 294, 379, 479, 595, 728, 879, 1049, 1239, 1450, 1683, 1939, 2219, 2524, 2855, 3214, 3602, 4020, 4469, 4950, 5464, 6012, 6595, 7214, 7870, 8564, 9297, 10070, 10884, 11740, 12639, 13582, 14570, 15604, 16685, 17815, 18995

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..45.

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) = 2*a(1) - a(0) + b(1) - 1 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294867 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A305329 A161212 A256058 * A033547 A050531 A290699

Adjacent sequences: A294864 A294865 A294866 * A294868 A294869 A294870

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved