A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), 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, 9, 20, 44, 91, 187, 379, 764, 1534, 3075, 6157, 12322, 24652, 49313, 98635, 197280, 394571, 789153, 1578318, 3156648, 6313309, 12626631, 25253276, 50506566, 101013147, 202026309

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.

Links

Table of n, a(n) for n=0..26.

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

Formula

a(n+1)/a(n) -> 2.

Example

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

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] = a[ n - 1] + 2 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}]  (* A295148 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295145, A295146, A295147.

Sequence in context: A011796 A164680 A210634 * A364535 A176163 A203861

Adjacent sequences: A295145 A295146 A295147 * A295149 A295150 A295151

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved