A295063 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 10, 21, 51, 97, 219, 405, 896, 1643, 3609, 6599, 14465, 26427, 57893, 105743, 231609, 423011, 926478, 1692089, 3705959, 6768405, 14823887, 27073673, 59295603, 108294749, 237182471

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 2.19... and 1.82... .

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

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 - 1] + 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}]  (* A295063 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A330273 A335666 A192033 * A298856 A006308 A008837

Adjacent sequences: A295060 A295061 A295062 * A295064 A295065 A295066

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved