A295062 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 6, 16, 29, 71, 124, 293, 506, 1183, 2036, 4745, 8158, 18995, 32649, 75998, 130615, 304012, 522481, 1216070, 2089947, 4864304, 8359813, 19457242, 33439279, 77828996, 133757146, 311316015

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 1.71... and 2.32... .

Links

Table of n, a(n) for n=0..27.

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

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = 4*a(0) + b(0) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=4;
a[n_] := a[n] = 4 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}]  (* A295062 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A151954 A032247 A052281 * A291611 A308401 A196261

Adjacent sequences: A295059 A295060 A295061 * A295063 A295064 A295065

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved