A294866 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 17, 33, 57, 90, 133, 187, 253, 332, 425, 533, 657, 799, 960, 1141, 1343, 1567, 1814, 2085, 2381, 2703, 3052, 3429, 3835, 4271, 4738, 5237, 5770, 6338, 6942, 7583, 8262, 8980, 9738, 10537, 11378, 12262, 13190, 14163, 15182, 16248, 17362, 18525, 19738

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) = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294866 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A086513 A166381 A083723 * A045947 A321123 A145066

Adjacent sequences: A294863 A294864 A294865 * A294867 A294868 A294869

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved