A324007 -id:A324007 - cp's OEIS frontend

A065563 Product of three consecutive Fibonacci numbers.

2, 6, 30, 120, 520, 2184, 9282, 39270, 166430, 704880, 2986128, 12649104, 53583010, 226980390, 961505790, 4073001576, 17253515288, 73087057560, 309601753890, 1311494059590, 5555578014142, 23533806080736, 99690802394400, 422297015565600, 1788878864806850, 7577812474550214

Offset: 1

Len Smiley, Nov 30 2001

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 89, No. 32, with a minus sign.

Harry J. Smith, Table of n, a(n) for n=1..200
V. E. Hoggatt and D. A. Lind, The Heights of Fibonacci Polynomials and an Associated Function, Fibonacci Quarterly, Vol. 5, No. 2 (April, 1967), pp. 141-152.
Joseph S. Ozbolt, A New Sequence Derived From a Combination of Cubes with Volume Fn^3, Fibonacci Quarterly, Vol. 50, No. 1 (2012), pp. 19-26.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A001655, A079586, A324007.

[&*[Fibonacci(n+k): k in [0..2] ]: n in [1..30]]; // Vincenzo Librandi, Apr 09 2020

with (combinat):a:=n->fibonacci(n)*fibonacci(n+1)*fibonacci(n+2): seq(a(n), n=1..22); # Zerinvary Lajos, Oct 07 2007

Times@@@Partition[Fibonacci[Range[30]],3,1] (* Harvey P. Dale, Aug 18 2011 *)

a(n) = { fibonacci(n)*fibonacci(n + 1)*fibonacci(n + 2) } \\ Harry J. Smith, Oct 22 2009

a(n) = A000045(n)*A000045(n+1)*A000045(n+2).
G.f.: 2/(1 - 3*x - 6*x^2 + 3*x^3 + x^4).
a(n) = 2*A001655(n).
a(n) = Fibonacci(n+1)^3-(-1)^n*Fibonacci(n+1). - Gary Detlefs, Feb 02 2011
This corrects a sign mistake in the Koshy reference. - Wolfdieter Lang, Aug 07 2012
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).
O.g.f.: 2*x/((1 + x - x^2)*(1 - 4*x - x^2)) (compare with A001655). - Wolfdieter Lang, Aug 06 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = A079586 - 3. - Amiram Eldar, Oct 04 2020
Sum_{n>=1} 1/a(n) = A324007. - Amiram Eldar, Feb 09 2023

Offset changed from 0 to 1 by Harry J. Smith, Oct 22 2009

A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).

Original entry on oeis.org

2, 8, 9, 1, 4, 4, 6, 4, 8, 5, 7, 0, 6, 7, 1, 5, 8, 3, 1, 1, 2, 3, 0, 5, 5, 0, 9, 6, 1, 5, 7, 2, 9, 1, 6, 6, 9, 5, 4, 8, 8, 1, 9, 5, 1, 5, 8, 9, 6, 9, 1, 3, 6, 0, 0, 2, 5, 0, 2, 6, 4, 8, 5, 0, 6, 3, 0, 3, 5, 7, 6, 1, 7, 3, 8, 8, 6, 4, 5, 5, 1, 5, 8, 2, 4, 1, 1, 5, 8, 3, 1, 8, 2, 8, 5

Offset: 0

Michel Lagneau, Mar 26 2011

André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)). - Amiram Eldar, Oct 30 2020

			0.2891446485706715831123055096157291669...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.

Cf. A000045, A001622, A079586.

Cf. A153386, A153387, A324007.

with(combinat, fibonacci):Digits:=100:s:=0:for n from 1 to 2000 do: a1:=fibonacci(n):s:=s+evalf(1/a1)*(-1)^(n+1):od:print(s):

digits = 95; NSum[(-1)^(n+1)*(1/Fibonacci[n]), {n, 1, Infinity}, WorkingPrecision -> digits+1, NSumTerms -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Jan 28 2014 *)

-sumalt(n=1,(-1)^n/fibonacci(n)) \\ Charles R Greathouse IV, Oct 03 2016

Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) + (-1)^k), where phi is the golden ratio (A001622). - Amiram Eldar, Oct 04 2020
Equals A153387 - A153386. - Joerg Arndt, Oct 04 2020
Equals 1 - A324007. - Amiram Eldar, Feb 09 2023

Offset corrected by Arkadiusz Wesolowski, Jun 28 2011

A324008 Decimal expansion of the sum of reciprocals of the products of k consecutive Fibonacci numbers, k>0.

Original entry on oeis.org

6, 0, 9, 2, 2, 4, 3, 4, 4, 7, 4, 3, 4, 4, 4, 2, 1, 1, 2, 1, 1, 7, 3, 8, 7, 3, 7, 6, 3, 0, 4, 0, 4, 1, 3, 1, 4, 6, 2, 7, 5, 2, 4, 3, 0, 7, 2, 8, 4, 7, 5, 5, 2, 3, 6, 7, 8, 8, 5, 5, 1, 4, 2, 9, 0, 3, 3, 5, 7, 9, 6, 5, 5, 2, 8, 6, 4, 6, 8, 1, 7, 4, 4, 5, 6, 8, 0, 7, 9, 1, 6, 7, 7, 3, 6, 1, 4, 5, 1, 4, 4, 7, 4, 6, 5

Offset: 1

Robert G. Wilson v, Feb 11 2019

Start of Array, the decimal expansion of Sum_{k>=1} 1/Product of the k consecutive Fibonacci numbers.
k A_xxxxx Expansion
1 A079586 3.3598856662431775531720113029189271796889051337319684864955538153251
2 A290565 1.7738775832851323438023627656769659228307232393594341108392290498649
3 A324007 0.7108553514293284168876944903842708330451180484103086399749735149369
4         0.2049150281252628794885329140859047056992270504855928446613784432368
5         0.0378540002823260756631035758318263246518410219564654534474085675610
6         0.0045033916811269635259578369635768898174496948588364334148979517071
7         0.0003362411268453457928115656517725694972839133469989715601437444837
8         0.0000157197618585596646075219438686990100758465336322798458353726393
9 A322711 0.0000004571522762064818372598445572889518549113726012557938158960751
...
          ---------------------------------------------------------------------
Total     6.0922434474344421121173873763040413146275243072847552367885514290335

			6.0922434474344421121173873763040413146275243072847552367885514290335796552...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000045, A079586, A290565, A322711, A324007.

f[n_] := Sum[ N[ 1/Product[ Fibonacci@j, {j, k, k +n -1}], 110], {k, 525}]; Sum[ f[n], {n, 35}]

cp's OEIS Frontend

A065563 Product of three consecutive Fibonacci numbers.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A324008 Decimal expansion of the sum of reciprocals of the products of k consecutive Fibonacci numbers, k>0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica