A294870 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 23, 45, 76, 117, 170, 236, 316, 411, 522, 650, 796, 961, 1146, 1352, 1580, 1831, 2106, 2407, 2735, 3091, 3476, 3891, 4337, 4815, 5326, 5871, 6451, 7067, 7720, 8411, 9141, 9911, 10722, 11575, 12471, 13411, 14396, 15427, 16506, 17634, 18812, 20041

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..44.

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, 10, 11, 12, 13, 14, ...)
m

Mathematica

ex := 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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294870 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A027702 A051897 A209294 * A032636 A101986 A023542

Adjacent sequences: A294867 A294868 A294869 * A294871 A294872 A294873

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved