A295617 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 5, 13, 29, 60, 115, 210, 370, 636, 1074, 1792, 2963, 4868, 7961, 12977, 21105, 34269, 55582, 90081, 145916, 236274, 382492, 619094, 1001941, 1621418, 2623772, 4245634, 6869882, 11116025, 17986450, 29103053, 47090117, 76193821, 123284627, 199479176

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295613 for a guide to related sequences.

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Table of n, a(n) for n=0..35.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Example

a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that
b(3) = 7 (least "new number")
a(3) = 2*a(2) - a(0) + b(1) = 13
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; b[2] = 6;
a[n_] := a[n] = 2 a[n - 1] - a[n - 3] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 30}]  (* A295617 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045, A295613.

Sequence in context: A110225 A065047 A282565 * A290113 A238216 A067932

Adjacent sequences: A295614 A295615 A295616 * A295618 A295619 A295620

Keyword

nonn,easy

Author

Clark Kimberling, Nov 25 2017

Status

approved