A295998 Solution of the complementary equation a(n) = 2*a(n-2) + 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, 5, 8, 16, 23, 41, 56, 93, 124, 199, 262, 413, 541, 844, 1101, 1708, 2223, 3438, 4470, 6901, 8966, 13829, 17960, 27687, 35950, 55405, 71932, 110843, 143898, 221721, 287832, 443479, 575702, 886997, 1151444, 1774036, 2302931, 3548116, 4605907, 7096278

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 1.298123759410105...

See A295860 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..999

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

Formula

a(0) = 1, a(1) = 2, b(0) = 3, so that a(2) = 5, b(1) = 4.

Complement: (b(n)) = (3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, ...)

Mathematica

mex[t_] := NestWhile[# + 1 &, 1, MemberQ[t, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 2] + b[n - 2];  (* A295998 *)
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];
Table[b[n], {n, 0, 30}]

Crossrefs

Cf. A001622, A000045, A294860.

Sequence in context: A361270 A360671 A168470 * A129299 A171238 A096541

Adjacent sequences: A295995 A295996 A295997 * A295999 A296000 A296001

Keyword

nonn,easy

Author

Clark Kimberling, Dec 02 2017

Status

approved