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A295141 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 5
1, 2, 8, 22, 57, 142, 348, 847, 2052, 4962, 11988, 28951, 69904, 168774, 407468, 983727, 2374940, 5733626, 13842212, 33418071, 80678377, 194774849, 470228100 (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..22.

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

FORMULA

a(n+1)/a(n) -> 1 + sqrt(2).

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4

a(2) =2*a(1) + a(0) + b(0) = 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; b[1] = 4;

a[n_] := a[n] = 2 a[ n - 1] + a[n - 2] + b[n - 2];

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 18}]  (* A295141 *)

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

CROSSREFS

Cf. A295053, A295142, A295143, A295144.

Sequence in context: A005803 A145654 A221880 * A074352 A301555 A261561

Adjacent sequences:  A295138 A295139 A295140 * A295142 A295143 A295144

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 19 2017

STATUS

approved

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Last modified November 20 20:28 EST 2019. Contains 329347 sequences. (Running on oeis4.)