A295064 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-1), 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, 14, 31, 48, 121, 258, 395, 980, 2077, 3175, 7856, 16633, 25418, 62867, 133084, 203365, 502958, 1064695, 1626944, 4023689, 8517586, 13015579, 32189540, 68140717, 104124662

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 1.52..., 2.11..., 2.47...

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(2) = 14
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, ...)

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 - 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}]  (* A295064 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A198785 A222380 A271867 * A052974 A284415 A318227

Adjacent sequences: A295061 A295062 A295063 * A295065 A295066 A295067

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved