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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A080084 A065093 A168470 * A129299 A171238 A096541

Adjacent sequences:  A295995 A295996 A295997 * A295999 A296000 A296001

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 02 2017

STATUS

approved

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Last modified March 20 21:49 EDT 2019. Contains 321352 sequences. (Running on oeis4.)