A295065 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 5, 12, 30, 47, 104, 249, 386, 843, 2005, 3102, 6759, 16056, 24833, 54090, 128467, 198684, 432741, 1027758, 1589495, 3461952, 8222089, 12715986, 27695643, 65776740, 101727917

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 three convergent subsequences, with limits 1.54..., 2.17..., 2.37...

Links

Table of n, a(n) for n=0..26.

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
a(3)  = 8*a(0) + b(1) = 12
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2;
a[n_] := a[n] = 8 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, 18}]  (* A295065 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295064.

Sequence in context: A056690 A317580 A066951 * A046091 A002905 A220832

Adjacent sequences: A295062 A295063 A295064 * A295066 A295067 A295068

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved