A295069 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-2) + n, where a(0) = 3, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 4, 7, 9, 13, 17, 24, 31, 45, 59, 86, 114, 168, 223, 331, 441, 657, 877, 1309, 1748, 2612, 3490, 5218, 6974, 10430, 13941, 20853, 27875, 41699, 55743, 83391, 111479, 166775, 222951, 333543, 445895, 667079, 891783, 1334150, 1783558, 2668292, 3567108

Offset

0,1

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 1.33..., 1.49...

Links

Table of n, a(n) for n=0..41.

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

Example

a(0) = 3, a(1) = 4, b(0) = 1
a(2) = 2*a(0) - b(0)  + 2 = 7
Complement: (b(n)) = (1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, ... )

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 4; b[0] = 1;
a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295069 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295068.

Sequence in context: A061568 A146994 A330146 * A349360 A103054 A140208

Adjacent sequences: A295066 A295067 A295068 * A295070 A295071 A295072

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved