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A295367 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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. 4
1, 2, 15, 37, 82, 161, 299, 532, 921, 1563, 2616, 4335, 7133, 11692, 19097, 31095, 50534, 82009, 132963, 215434, 348903, 564889, 914392, 1479931, 2395025, 3875712, 6271549, 10148131, 16420610, 26569733, 42991399, 69562254, 112554843 (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 A295357 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..32.

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

FORMULA

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

EXAMPLE

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

b(2) = 5 (least "new number")

a(2) = a(1) + a(0) + b(1)*b(0) = 15

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, ...)

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] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2];

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

z = 32; u = Table[a[n], {n, 0, z}]   (* A295367 *)

v = Table[b[n], {n, 0, 10}]  (* complement *)

CROSSREFS

Cf. A001622, A295357.

Sequence in context: A075542 A064113 A007217 * A214541 A180223 A070009

Adjacent sequences:  A295364 A295365 A295366 * A295368 A295369 A295370

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 21 2017

STATUS

approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)