A294863 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 9, 15, 18, 26, 31, 40, 46, 56, 63, 75, 83, 97, 106, 121, 131, 147, 158, 175, 188, 206, 220, 239, 255, 275, 292, 313, 331, 353, 372, 395, 416, 440, 462, 487, 510, 537, 561, 589, 614, 643, 669, 699, 726, 757, 786, 818, 848, 881, 912, 946, 979, 1014

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..54.

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) = a(0) + b(0) + 3 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + b[n - 2] + 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}]  (* A294863 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860, A294864.

Sequence in context: A226824 A168132 A211280 * A085544 A154789 A106352

Adjacent sequences: A294860 A294861 A294862 * A294864 A294865 A294866

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved