A295068 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-1) + n, where a(0) = 4, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

4, 5, 8, 10, 14, 18, 25, 32, 46, 60, 87, 115, 169, 224, 332, 442, 658, 878, 1310, 1749, 2613, 3491, 5219, 6975, 10431, 13942, 20854, 27876, 41700, 55744, 83392, 111480, 166776, 222952, 333544, 445896, 667080, 891784, 1334151, 1783559, 2668293, 3567109

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) = 4, a(1) = 5, b(0) = 1
a(2) = 2*a(0) - b(1)  + 2 = 8
Complement: (b(n)) = (1, 2, 3, 6, 7, 9, 11, 12, 13, 15, 16, 17, 19, ... )

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 4; a[1] = 5; 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}]  (* A295068 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295069.

Sequence in context: A140459 A319515 A139132 * A022435 A190394 A116050

Adjacent sequences: A295065 A295066 A295067 * A295069 A295070 A295071

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved