A295139 Solution of the complementary equation a(n) = 3*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 10, 23, 37, 77, 120, 242, 372, 739, 1130, 2232, 3406, 6713, 10236, 20158, 30728, 60495, 92206, 181509, 276643, 544553, 829956, 1633687, 2489897, 4901091, 7469722, 14703305, 22409199, 44109949, 67227632, 132329883

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 1.52..., 1.96...

Links

Table of n, a(n) for n=0..32.

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1]=4;
a[n_] := a[n] = 3 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}]  (* A295139 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A200572 A342136 A049750 * A134016 A072297 A358908

Adjacent sequences: A295136 A295137 A295138 * A295140 A295141 A295142

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved