A295138 Solution of the complementary equation a(n) = 3*a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 11, 27, 41, 90, 133, 282, 412, 860, 1251, 2596, 3770, 7806, 11329, 23438, 34008, 70336, 102047, 211032, 306166, 633122, 918526, 1899395, 2755608, 5698216, 8266856, 17094681, 24800602, 51284078, 74401842

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 1.45..., 2.06...

Links

Table of n, a(n) for n=0..31.

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 2] + 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}]  (* A295138 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A201630 A023862 A024479 * A023864 A024857 A024481

Adjacent sequences: A295135 A295136 A295137 * A295139 A295140 A295141

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved