A194159 -id:A194159 - cp's OEIS frontend

A062073 Decimal expansion of Fibonacci factorial constant.

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

Offset: 1

Jason Earls, Jun 27 2001

The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.

Two closely related constants are A194159 and A194160. [Johannes W. Meijer, Aug 21 2011]

			1.226742010720353244417630230455361655871409690440250419643297301214...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.

Harry J. Smith, Table of n, a(n) for n=1..5000
M. Griffiths, Symmetric rational expressions in the Fibonacci numbers, Fib. Q., 46/47 (2008/2009), 262-267. [_N. J. A. Sloane_, Dec 05 2009]
Simon Plouffe, Fibonacci factorials
Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Cf. A003266, A003267, A003268, A056569, A062072, A062381, A135407, A181926.

Cf. A218490, A253924, A256831, A259314, A259405.

RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* Alonso del Arte, Dec 20 2010 *)
RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1,0,-I/GoldenRatio]^(1/3)], 120]][[1]] (* Vaclav Kotesovec, Jul 19 2015, after Eric W. Weisstein *)

\p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1,17000,(1-a^n))

{ default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ Harry J. Smith, Jul 31 2009

C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
C = QPochhammer[ -1/GoldenRatio^2]. [Eric W. Weisstein, Dec 01 2009]
C = A194159 * A194160. [Johannes W. Meijer, Aug 21 2011]
C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - Vaclav Kotesovec, Jun 08 2013
C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - Vladimir Reshetnikov, Sep 20 2016

A152749 a(n) = (n+1)(3n+1)/4 for n odd, a(n) = n(3n+2)/4 for n even.

Original entry on oeis.org

0, 2, 4, 10, 14, 24, 30, 44, 52, 70, 80, 102, 114, 140, 154, 184, 200, 234, 252, 290, 310, 352, 374, 420, 444, 494, 520, 574, 602, 660, 690, 752, 784, 850, 884, 954, 990, 1064, 1102, 1180, 1220, 1302, 1344, 1430, 1474, 1564, 1610, 1704, 1752, 1850, 1900, 2002

Offset: 0

Vincenzo Librandi, Dec 31 2009

Interleaving of A049450 and A049451 (for n > 0).
Also, integer values of k*(k+1)/3. - Charles R Greathouse IV, Dec 11 2010
The nonzero coefficients of the expansion of f(a) = Product_{k>=1} (1-a^(2k)), see A194159, occur at the terms of the sequence given above, i.e., f(a) = 1 - a^2 - a^4 + a^10 + a^14 - a^24 - a^30 + a^44 + a^52 - a^70 - a^80 + ... = Sum_{n>=0} (-1)^binomial(n+1,2)*a^A152749(n). - Johannes W. Meijer, Aug 21 2011
Partial sums of A109043. - Reinhard Zumkeller, Mar 31 2012
Nonnegative k such that 12*k+1 is a square. - Vicente Izquierdo Gomez, Jul 22 2013
Equivalently, numbers of the form h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the fifth comment of A062717). - Bruno Berselli, Feb 02 2017
For n > 0, a(n-1) is the sum of the largest parts of the partitions of 2n into two even parts. - Wesley Ivan Hurt, Dec 19 2017
The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)/2) * Product_{k >= n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - -  .... - Peter Bala, Dec 15 2024
Sequence terms occur as exponents in the expansions of Sum_{n >= 0} q^(n*(2*n+1)) * Product_{k >= 2*n+2} 1 - q^k  = Sum_{n >= 0} q^(n*(2*n-1)) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Jun 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A049450 (n*(3*n-1)), A049451 (n*(3*n+1)), A153383 (12n+1 is not prime).

a152749 n = a152749_list !! (n-1)
a152749_list = scanl1 (+) a109043_list
-- Reinhard Zumkeller, Mar 31 2012

[IsOdd(n) select (n+1)*(3*n+1)/4 else n*(3*n+2)/4: n in [0..52]];

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // Bruno Berselli, Nov 13 2012

A152749 := proc(n): if type(n,even) then n*(3*n+2)/4  else (n+1)*(3*n+1)/4 fi: end: seq(A152749(n), n=0..51); # Johannes W. Meijer, Aug 21 2011

Table[If[OddQ[n],(n+1)*(3*n+1)/4,n*(3*n+2)/4],{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
LinearRecurrence[{1,2,-2,-1,1}, {0, 2, 4, 10, 14}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
Select[Range[1,1000], IntegerQ[Sqrt[12#+1]]&] (* Vicente Izquierdo Gomez, Jul 22 2013 *)

From R. J. Mathar, Jan 03-06 2009: (Start)
G.f.: 2*x*(1+x+x^2)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) = A003154(n+1)/8 - (-1)^n*A005408(n)/8.
a(n) = 2*A001318(n) = ((6*n^2+6*n+1) - (2*n+1)*(-1)^n)/8. (End)
From Amiram Eldar, Mar 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(log(3)-1). (End)

Edited, typo corrected and extended by Klaus Brockhaus, Jan 02 2009

Leading term a(0)=0 added by Johannes W. Meijer, Aug 21 2011

A194157 Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

Original entry on oeis.org

1, 3, 24, 504, 27720, 3991680, 1504863360, 1485300136320, 3838015552250880, 25964175210977203200, 459851507161617245875200, 21322394684069868456741273600, 2588389457883293541569193426124800, 822618641999347403739646931950148812800

Offset: 1

Johannes W. Meijer, Aug 21 2011

The terms of this sequence are Fibonacci double factorial numbers.
a(n) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio. For the decimal expansion of C2 see A194159.
Product of first n terms of the binomial transform of the Fibonacci numbers. - Vaclav Kotesovec, Oct 29 2017

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

Vincenzo Librandi, Table of n, a(n) for n = 1..70
Eric Weisstein, Fibonorial, Mathworld.

Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.

[&*[Fibonacci(2*i): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Sep 15 2016

with(combinat): A194157 :=proc(n): mul(fibonacci(2*i), i=1..n) end: seq(A194157(n), n=1..14);

FoldList[Times, Fibonacci[2 Range[20]]] (* or *)
Table[Round[GoldenRatio^(n(n-1)) QFactorial[n, 1/GoldenRatio^4]], {n, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
Table[Product[Sum[Binomial[m, k]*Fibonacci[k], {k, 1, m}], {m, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Oct 29 2017 *)

{a(n) = if( n<0, 0, prod(k=1, n, fibonacci(2*k)))}; /* Michael Somos, Oct 06 2014 */

a(n) = Product_{i=1..n} F(2*i) with F(n) = A000045(n).
a(n) = A123029(2*n).
a(n+1)/a(n) = A001906(n+1).
0 = a(n)*(3*a(n+2)^2 - a(n+1)*a(n+3)) -a(n+1)^2*a(n+2) for all n>=0. - Michael Somos, Oct 06 2014

A194158 Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).

Original entry on oeis.org

1, 2, 10, 130, 4420, 393380, 91657540, 55911099400, 89290025741800, 373321597626465800, 4086378207619294646800, 117103340295746126693347600, 8785678105688353155168403690000, 1725665322163094950031867515982420000, 887387152950606153059937200876123854180000

Offset: 1

Johannes W. Meijer, Aug 21 2011

The terms of this sequence are Fibonacci double factorial numbers.

The a(n) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio A001622. For the decimal expansion of C1 see A194160.

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 130*x^4 + 4420*x^5 + 393380*x^6 + 91657540*x^7 + ...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

G. C. Greubel, Table of n, a(n) for n = 1..69
Eric Weisstein's World of Mathematics, Fibonorial

Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.

with(combinat): A194158 :=proc(n): mul(fibonacci(2*i-1), i=1..n) end: seq(A194158(n), n=1..15);

Table[Product[Fibonacci[2*k - 1], {k, 1, n}], {n, 1, 30}] (* G. C. Greubel, Aug 13 2018 *)

{a(n) = if( n<0, 1 / a(-n), prod(k=1, n, fibonacci(2*k - 1)))}; /* Michael Somos, Oct 07 2014 */

a(n) = Product_{i=1..n} F(2*i-1), where F(n) = A000045(n).
a(n) = A123029(2*n-1).
a(n+1)/a(n) = A001519(n+1).
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 07 2014
a(-n) = 1/a(n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1)*a(n+3) - 3*a(n+2)^2) + a(n+2)*(+a(n+1)^2) for all n in Z. - Michael Somos, Oct 07 2014
(F(1) + i)(F(3) + i)...(F(2n+1) + i) = a(n)(1 + F(2n+2)i) and (F(2n+1) + i)(1 + F(2n)i) = F(2n-1)(1 + F(2n+2)i) for all n in Z. - Michael Somos, Sep 16 2023

A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Aug 21 2011

A194158(n) = prod(i=1..n, F(2*i-1) ) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 and F(n) = A000045(n). The decimal expansion of the constant C1 is given here.

			C1 = 1.4728859290995693146071...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

Eric Weisstein, Fibonorial, Mathworld.

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

RealDigits[Product[1-((-1)/GoldenRatio^2)^(2k-1),{k,1000}],10, 100] [[1]] (* Harvey P. Dale, Aug 28 2011 *)
RealDigits[QPochhammer[-GoldenRatio^2, 1/GoldenRatio^4]/(GoldenRatio Sqrt[5]), 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

C1 = prod(k>=1, 1-alpha^(2*k-1) ) where alpha = (-1/phi^2) and phi = (1+sqrt(5))/2.

C1 = A062073/A194159

A349272 a(n) = Product_{k = 1..2n+1} Fibonacci(2k) / Sum_{k = 1..2n+1} Fibonacci(2k).

Original entry on oeis.org

1, 2, 315, 2471040, 918185538816, 16047302734562299200, 13178031727820369629763174400, 508406658175888466343652105865846784000, 921456090985190879093613420564815806955580862464000, 78458394721620642094151397745899367347021362840662985785265356800

Offset: 0

Peter Bala, Nov 12 2021

Let m be an even positive integer. We conjecture that the sequence defined by b_m(n) = Product_{k = 1..2*n+1} Fibonacci(m*k) / Sum_{k = 1..2*n+1} Fibonacci(m*k) is integral. The formula given below proves the conjecture in the present case m = 2. The cases m = 4 and m = 6 of the conjecture can be proved in a similar manner.

More generally, if F(n,x) denotes the n-th Fibonacci polynomial we conjecture that, for each n, the rational function Product_{k = 1..2*n+1} F(m*k,x) / Sum_{k = 1..2*n+1} F(m*k,x) is an integral polynomial.

Wikipedia, Fibonacci polynomials

Cf. A000045, A159951, A175553.

Cf. A003266, A194157.

with(combinat):
seq(mul(fibonacci(2*k), k = 1..2*n+1)/add(fibonacci(2*k), k = 1..2*n+1), n = 0..10);

Table[Product[ Fibonacci[2k],{k,2n+1}]/Sum[Fibonacci[2k],{k,2n+1}],{n,0,9}] (* Stefano Spezia, Nov 13 2021 *)

a(n) = prod(k = 1, 2*n+1, fibonacci(2*k)) / sum(k = 1, 2*n+1, fibonacci(2*k)); \\ Michel Marcus, Nov 12 2021

a(n) = F(2*n+1)/F(2*n+2) * Product_{k = 1..2*n} Fibonacci(2*k), shows a(n) to be integral. Cf. A159951.

a(n) ~ A194159 * phi^(4*n^2 + 2*n - 1) / 5^n, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 31 2023

cp's OEIS Frontend

A062073 Decimal expansion of Fibonacci factorial constant.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A152749 a(n) = (n+1)*(3*n+1)/4 for n odd, a(n) = n*(3*n+2)/4 for n even.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

Formula

Extensions

A194157 Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A194158 Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A349272 a(n) = Product_{k = 1..2*n+1} Fibonacci(2*k) / Sum_{k = 1..2*n+1} Fibonacci(2*k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A152749 a(n) = (n+1)(3n+1)/4 for n odd, a(n) = n(3n+2)/4 for n even.

A349272 a(n) = Product_{k = 1..2n+1} Fibonacci(2k) / Sum_{k = 1..2n+1} Fibonacci(2k).