A295067 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 4, 11, 14, 29, 36, 67, 82, 146, 177, 307, 370, 631, 758, 1281, 1536, 2583, 3094, 5189, 6212, 10403, 12450, 20833, 24928, 41696, 49887, 83424, 99807, 166882, 199649, 333801, 399336, 667641, 798712, 1335323, 1597466, 2670689, 3194976, 5341423, 6389998

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 1.19..., 1.67...

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=5;
a[n_] := a[n] = 2 a[n - 2] + 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, 18}]  (* A295067 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295066.

Sequence in context: A133621 A216558 A049619 * A263262 A249650 A114951

Adjacent sequences: A295064 A295065 A295066 * A295068 A295069 A295070

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved