A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

P. Majer proved that a(n)/n -> 4, that (a(n) - 4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964. - Clark Kimberling, Feb 10 2018

Links

Peter J. C. Moses, Table of n, a(n) for n = 0..9999

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

Pietro Majer, Limit associated with complementary sequences Math Overflow.

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0)*b(1) + a(1)*b(0) = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 500}];  (* A296220 *)
Table[b[n], {n, 0, 20}]

Crossrefs

Cf. A296000.

Sequence in context: A177856 A343476 A343477 * A297835 A298000 A058216

Adjacent sequences: A296217 A296218 A296219 * A296221 A296222 A296223

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved