A295998 - OEIS

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

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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}]]];