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A295061 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 3, 8, 17, 38, 75, 161, 310, 655, 1252, 2633, 5022, 10547, 20104, 42206, 80435, 168844, 321761, 675398, 1287067, 2701616, 5148293, 10806490, 20593199, 43225988, 82372825, 172903982 (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.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.09... and 2.09... .

LINKS

Table of n, a(n) for n=0..26.

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

EXAMPLE

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

a(2) = 4*a(0) + b(1) = 8

Complement: (b(n)) = (2, 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] = 3; b[0] = 2;

a[n_] := a[n] = 4 a[n - 2] + b[n - 1];

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

Table[a[n], {n, 0, 18}]  (* A295061 *)

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

CROSSREFS

Cf. A295053.

Sequence in context: A097391 A202554 A034481 * A247374 A336512 A046994

Adjacent sequences:  A295058 A295059 A295060 * A295062 A295063 A295064

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 18 2017

STATUS

approved

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Last modified May 7 06:18 EDT 2021. Contains 343636 sequences. (Running on oeis4.)