A218490 -id:A218490 - 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

A070825 One half of product of first n+1 Lucas numbers A000032.

Original entry on oeis.org

1, 1, 3, 12, 84, 924, 16632, 482328, 22669416, 1722875616, 211913700768, 42170826452832, 13579006117811904, 7074662187380001984, 5963940223961341672512, 8134814465483270041306368, 17953535525321576981163154176, 64112075360923351399733623562496, 370439571435415124387660876944101888

Offset: 0

Wolfdieter Lang, May 10 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..90
R. Grünwald, E. Heidel, A. Strätz, M. Sünkel and R. Terbach, Induction on Number Series, Fakultät fur Wirtschaftsinformatik und Angewandte Informatik, Otto-Friedrich-Universität Bamberg, 2012. - _N. J. A. Sloane_, Feb 07 2013

Cf. A000032, A003266 (for Fibonacci), A003046 (for Catalan), A101690, A135407, A218490.

[1] cat [&*[Lucas(i+1): i in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 15 2016

c := arccsch(2) - I*Pi/2:
A070825 := n -> local j; 2^n*mul(I^j*cosh(c*j), j = 1..n):
seq(simplify(A070825(n)), n = 0..18);  # Peter Luschny, Jul 07 2025

FoldList[Times, LucasL[Range[0, 20]]]/2 (* or *)
Table[Round[GoldenRatio^(n(n+1)/2) QPochhammer[-1, GoldenRatio-2, n+1]]/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

a(n) = prod(k=0, n, fibonacci(k+1)+fibonacci(k-1))/2; \\ Michel Marcus, Mar 18 2016

a(n) = (Product_{k=0..n} L(k))/2 with L = A000032.
Sum_{n>=0} 1/a(n) = 1 + A101690. - Amiram Eldar, Nov 09 2020
a(n) = 2^n*Product_{j=1..n} i^j*cosh(c*j), where c = arccsch(2) - i*Pi/2. - Peter Luschny, Jul 07 2025

A135407 Partial products of A000032 (Lucas numbers beginning at 2).

Original entry on oeis.org

2, 2, 6, 24, 168, 1848, 33264, 964656, 45338832, 3445751232, 423827401536, 84341652905664, 27158012235623808, 14149324374760003968, 11927880447922683345024, 16269628930966540082612736

Offset: 0

Jonathan Vos Post, Dec 09 2007

This is to A000032 as A003266 is to A000045. a(n) is asymptotic to C*phi^(n*(n+1)/2) where phi=(1+sqrt(5))/2 is the golden ratio and C = 1.3578784076121057013874397... (see A218490). - Corrected and extended by Vaclav Kotesovec, Oct 30 2012

			a(0) = L(0) = 2.
a(1) = L(0)*L(1) = 2*1 = 2.
a(2) = L(0)*L(1)*L(2) = 2*1*3 = 6.
a(3) = L(0)*L(1)*L(2)*L(3) = 2*1*3*4 = 24.

G. C. Greubel, Table of n, a(n) for n = 0..95

Cf. A000032, A000045, A000204, A003266, A070825, A218490.

Rest[FoldList[Times,1,LucasL[Range[0,20]]]] (* Harvey P. Dale, Aug 21 2013 *)
Table[Round[GoldenRatio^(n(n+1)/2) QPochhammer[-1, GoldenRatio-2, n+1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 14 2016 *)

a(n) = prod(k=0, n, fibonacci(k+1)+fibonacci(k-1)); \\ Michel Marcus, Oct 13 2016

a(n) = Product_{k=0..n} A000032(k).

C = exp( Sum_{k>=1} 1/(k*(((3-sqrt(5))/2)^k-(-1)^k)) ). - Vaclav Kotesovec, Jun 08 2013

A256831 Decimal expansion of Pell factorial constant.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 10 2015

			1.141982569667791206028043338367860150864730482408540791556...

Cf. A000129, A099929, A256832.

Cf. A062073, A218490, A253924.

RealDigits[N[QPochhammer[2*Sqrt[2]-3], 105]][[1]]

Equals limit n->infinity A256832(n) / ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)).

A259314 Decimal expansion of partition factorial constant.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 24 2015

			0.91101673133224995186154746959468345278073860978008093028132149022759...

Vaclav Kotesovec, The partition factorial constant and asymptotics of the sequence A058694

Cf. A000041, A058694, A062073, A218490, A253924, A256831, A259405.

(* The iteration cycle: *) Do[Print[Product[N[PartitionsP[k]/((E^(Sqrt[2/3]*Sqrt[k-1/24]*Pi) * (1 - Sqrt[3/2]/(Sqrt[k-1/24]*Pi))) / (4*Sqrt[3]*(k-1/24))), 150], {k, 1, n}]], {n, 500, 50000, 500}]

Equals limit n->infinity Product_{k=1..n} p(k) / (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where p(k) is the partition function A000041.

A186269 a(n) = Product_{k=0..n-1} A084057(k+1).

Original entry on oeis.org

1, 1, 6, 96, 5376, 946176, 544997376, 1011515129856, 6085275021213696, 118395110812733669376, 7456050498542715562622976, 1519364146391040406489059557376, 1001953802522449942301649259468947456, 2138185445843748536070796346094885374263296, 14766000790292725890315725371457440731168428261376

Offset: 0

Paul Barry, Feb 16 2011

a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), 2^j*F(j+1), 2^i*F(i+1)))_{0<=i,j<=n}.

			a(2)=6 since det[1, 1, 1; 1, 2, 2; 1, 2, 8]=6.

Cf. A000032, A070825, A135407, A218490.

Table[FullSimplify[Product[(1+Sqrt[5])^k/2 + (1-Sqrt[5])^k/2,{k,0,n}]],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)
Table[Product[LucasL[k]*2^(k-1),{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n) = Product_{k=0..n} (1+sqrt(5))^k/2+(1-sqrt(5))^k/2.
a(n) = Product_{k=0..n} Sum_{j=0..floor(k/2)} binomial(k,2*j)*5^j. [corrected by Jason Yuen, Feb 12 2025]
a(n) ~ c * (1+sqrt(5))^(n*(n+1)/2) / 2^(n+1), where c = A218490 = 1.3578784076121057013874397... is the Lucas factorial constant. - Vaclav Kotesovec, Jul 11 2015

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A062073 Decimal expansion of Fibonacci factorial constant.

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A070825 One half of product of first n+1 Lucas numbers A000032.

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A135407 Partial products of A000032 (Lucas numbers beginning at 2).

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A256831 Decimal expansion of Pell factorial constant.

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A259314 Decimal expansion of partition factorial constant.

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A186269 a(n) = Product_{k=0..n-1} A084057(k+1).

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