A294868 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, 5, 12, 24, 42, 67, 100, 142, 195, 260, 338, 430, 537, 660, 800, 958, 1135, 1332, 1550, 1791, 2056, 2346, 2662, 3005, 3376, 3776, 4206, 4667, 5160, 5686, 6246, 6841, 7472, 8140, 8846, 9591, 10377, 11205, 12076, 12991, 13951, 14957, 16010, 17111, 18261

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) - 2 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, ...)

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] - 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}]  (* A294868 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A127787 A116733 A116721 * A192981 A116713 A261369

Adjacent sequences: A294865 A294866 A294867 * A294869 A294870 A294871

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved