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A295137 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 8, 26, 80, 242, 729, 2190, 6573, 19722, 59169, 177510, 532533, 1597602, 4792809, 14378430, 43135293, 129405882 (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. See A295053 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..17.

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

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3

a(2) =3*a(1) + b(1) - n = 8

Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... )

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

a[0] = 1; a[1] = 2; b[0] = 3;

a[n_] := a[n] = 3 a[n - 1] + 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}]  (* A295137 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

Cf. A295053.

Sequence in context: A279735 A103453 A024023 * A126966 A002930 A060410

Adjacent sequences:  A295134 A295135 A295136 * A295138 A295139 A295140

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 19 2017

STATUS

approved

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Last modified October 18 23:44 EDT 2021. Contains 348071 sequences. (Running on oeis4.)