A294872 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 24, 49, 86, 137, 205, 292, 400, 531, 687, 870, 1082, 1325, 1601, 1912, 2260, 2647, 3075, 3546, 4063, 4628, 5243, 5910, 6631, 7408, 8243, 9138, 10095, 11116, 12203, 13358, 14583, 15880, 17251, 18698, 20223, 21828, 23515, 25286, 27143, 29088, 31123

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..43.

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) + 2 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)

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] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294872 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A204556 A185669 A360518 * A006002 A259969 A023662

Adjacent sequences: A294869 A294870 A294871 * A294873 A294874 A294875

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved