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A064170 a(1) = 1; a(n+1) = product of numerator and denominator in Sum_{k=1..n} 1/a(k). 4
1, 1, 2, 10, 65, 442, 3026, 20737, 142130, 974170, 6677057, 45765226, 313679522, 2149991425, 14736260450, 101003831722, 692290561601, 4745030099482, 32522920134770, 222915410843905, 1527884955772562, 10472279279564026, 71778070001175617, 491974210728665290 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The numerator and denominator in the definition have no common divisors >1.

For n >= 3, a(n) = Fibonacci(2*n-5)*Fibonacci(2*n-3). - Barry Cipra, Jun 06 2002

Also denominators in a system of Egyptian fraction for ratios of consecutive Fibonacci numbers: 1/2 = 1/2, 3/5 = 1/2 + 1/10, 8/13 = 1/2 + 1/10 + 1/65, 21/34 = 1/2 + 1/10 + 1/65 + 1/442 etc. (Rossi and Tout). - Barry Cipra, Jun 06 2002

a(n)-1 is a square. - Sture Sjöstedt, Nov 04 2011

REFERENCES

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427; http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423

Rossi and Tout, Historia Mathematica, vol. 29 (2002), 101-113.

LINKS

Table of n, a(n) for n=1..24.

C. Rossi and C. A. Tout, Were the Fibonacci Series and the Golden Section Known in Ancient Egypt?, Historia Mathematica, vol. 29 (2002), 101-113.

Index entries for linear recurrences with constant coefficients, signature (8, -8, 1).

FORMULA

2/(1+sqrt(5)) = 0.6180339... = 1/2 + 1/10 + 1/65 + 1/442 + ... = Sum_{n>=3} 1/a(n). - Gary W. Adamson, Jun 07 2003

Conjecture: a(n) = 8*a(n-1)-8*a(n-2)+a(n-3), n>4. G.f.: -x*(2*x^2+x^3-7*x+1)/((x-1)*(x^2-7*x+1)). - R. J. Mathar, Jul 03 2009

a(n+1) = (A005248(n)^2 - A001906(n)^2)/4, for n => 0. - Richard R. Forberg, Sep 05 2013

EXAMPLE

1/a(1) + 1/a(2) + 1/a(3) + 1/a(4) = 1 + 1 + 1/2 + 1/10 = 13/5. So a(5) = 13 * 5 = 65.

MATHEMATICA

A064170[1] := 1; A064170[n_] := A064170[n] = Module[{temp = Sum[1/A064170[i], {i, n - 1}]}, Numerator[temp] Denominator[temp]]; Table[A064170[n], {n, 20}](* Alonso del Arte, Sep 05 2013 *)

Join[{1}, LinearRecurrence[{8, -8, 1}, {1, 2, 10}, 23]] (* Jean-François Alcover, Sep 22 2017 *)

CROSSREFS

Cf. A000045, A059929, A058038.

Cf. A033890 (first differences). - R. J. Mathar, Jul 03 2009

Cf. A001906.

Sequence in context: A223127 A130721 A167449 * A151410 A230050 A278459

Adjacent sequences:  A064167 A064168 A064169 * A064171 A064172 A064173

KEYWORD

nonn

AUTHOR

Leroy Quet, Sep 19 2001

STATUS

approved

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Last modified January 18 18:56 EST 2018. Contains 297864 sequences.