A295066 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 6, 11, 19, 30, 47, 70, 106, 153, 226, 321, 468, 659, 954, 1338, 1929, 2698, 3881, 5420, 7787, 10866, 15601, 21760, 31231, 43551, 62494, 87135, 125022, 174305, 250080, 348647, 500198, 697333, 1000436, 1394707, 2000914, 2789457, 4001872, 5578959, 8003790

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.43..., 1.39... .

Links

Table of n, a(n) for n=0..40.

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

Example

a(0) = 1, a(1) = 3, a(2) = 2, b(0) = 2, b(1) = 4,
a(2) = 2*a(0) + b(1) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = 2 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}]  (* A295066 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295067.

Sequence in context: A332446 A116100 A358300 * A004133 A180415 A344003

Adjacent sequences: A295063 A295064 A295065 * A295067 A295068 A295069

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved