A294871 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 26, 51, 86, 132, 190, 262, 349, 452, 572, 710, 867, 1044, 1242, 1462, 1705, 1972, 2264, 2582, 2927, 3300, 3703, 4137, 4603, 5102, 5635, 6203, 6807, 7448, 8127, 8845, 9603, 10402, 11243, 12127, 13055, 14028, 15047, 16113, 17227, 18390, 19603, 20867

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) + 3 = 10
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] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294871 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A374180 A069894 A045605 * A212969 A277712 A009307

Adjacent sequences: A294868 A294869 A294870 * A294872 A294873 A294874

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved