A135407 -id:A135407 - 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

A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

Original entry on oeis.org

3, 6, 24, 120, 960, 11520, 218880, 6566400, 315187200, 24269414400, 3009407385600, 601881477120000, 194407717109760000, 101480828331294720000, 85649819111612743680000, 116912003087351395123200000, 258141702816871880432025600000

Offset: 0

Clark Kimberling, Jul 25 2024

nonn

Cf. A000032, A000142, A003266, A070825, A082480, A135407, A374655.

w[n_] := Product[LucasL[k] + 1, {k, 0, n}]
Table[w[n], {n, 0, 20}]

a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

A218490 Decimal expansion of Lucas factorial constant.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 30 2012

The Lucas factorial constant is associated with the Lucas factorial A135407.

			1.35787840761210570138743970976060718557860586529567870449687825438407191103...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A062073, A135407, A070825, A003266, A000032, A000045, A186269, A001622.

RealDigits[QPochhammer[-1, -1/GoldenRatio^2], 10, 105][[1]] (* slightly modified by Robert G. Wilson v, Dec 21 2017 *)

prodinf(j=0, 1 + ((sqrt(5) - 3)/2)^j) \\ Iain Fox, Dec 21 2017

Equals exp( Sum_{k>=1} 1/(k*(((3-sqrt(5))/2)^k-(-1)^k)) ). - Vaclav Kotesovec, Jun 08 2013
Equals Product_{k=0..infinity} (1 + (-1)^k/phi^(2*k)). - G. C. Greubel, Dec 23 2017
Equals lim_{n->oo} A135407(n)/phi^(n*(n+1)/2), where phi is the golden ratio (A001622). - Amiram Eldar, Jan 23 2022

A256832 Product of first n Pell numbers Pell(1), ... , Pell(n).

Original entry on oeis.org

1, 2, 10, 120, 3480, 243600, 41168400, 16796707200, 16544756592000, 39343431175776000, 225870638380130016000, 3130567047948602021760000, 104751903991408172250111360000, 8462068308233934970708495883520000, 1650314871813323167662424409683488000000

Offset: 1

Vaclav Kotesovec, Apr 10 2015

nonn

Cf. A000129, A256831.

Cf. A003266, A135407, A126772.

Table[Product[Expand[((1+Sqrt[2])^k-(1-Sqrt[2])^k)/(2*Sqrt[2])],{k,1,n}],{n,1,20}]
FoldList[Times,LinearRecurrence[{2,1},{1,2},20]] (* Harvey P. Dale, Oct 07 2015 *)
FoldList[Times, Fibonacci[Range[20], 2]] (* or *)
Table[Round[(1+Sqrt[2])^((n-1)n/2) QFactorial[n, Sqrt[8]-3]], {n, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

a(n)=my(q=quadgen(8)+1,Q=q); prod(k=2,n, imag(Q*=q)) \\ Charles R Greathouse IV, Feb 14 2022

a(n) = Product_{k=1..n} A000129(k).

a(n) ~ c * ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)), where c = A256831 = 1.1419825696677912... . - Vaclav Kotesovec, Apr 10 2015

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

A254231 Product of tribonacci numbers A000073(2) * ... * A000073(n).

Original entry on oeis.org

1, 1, 2, 8, 56, 728, 17472, 768768, 62270208, 9278260992, 2542243511808, 1281290729951232, 1187756506664792064, 2025124843863470469120, 6350791510355843391160320, 36631365431732504680212725760, 388622155865250142152376807587840

Offset: 2

Vaclav Kotesovec, Jan 27 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..89

Cf. A000073, A003266, A058265, A126772, A135407, A254232.

Table[Product[SeriesCoefficient[x^2/(1-x-x^2-x^3),{x,0,k}],{k,2,n}], {n,2,20}]

A254231_list, a, b, c, d = [1], 0, 0, 1, 1
for _ in range(15):
    a, b, c = b, c, a+b+c
    d *= c
    A254231_list.append(d) # Chai Wah Wu, Jan 27 2015

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = A058265 = 1.839286755214161132551852564653286600424178746097592246778758639404203222... is the root of the equation r^3 - r^2 - r - 1 = 0, d = 0.061463687669952618841340986526101395138659648898940720192319213600612851... is the root of the equation -1 + 36*d - 440*d^2 + 1936*d^3 = 0, c = 4.156714772910304733054135311449211887936035199917470476143821433373978333... .

A203009 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(0)=2.

Original entry on oeis.org

1, 3, 11, 50, 374, 4282, 78924, 2322060, 110101476, 8413051008, 1038251025216, 207035781419520, 66749863269991104, 34803836775900988992, 29353783726459293724224, 40050488883338399323186560, 88407698594458813846355350656

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the A000032(j), j=0..n-1, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 2
1 3 2
1 6 11 6
1 10 35 50 24
1 17 105 295 374 168
1 28 292 1450 3619 4282 1848
1 46 796 6706 29719 69424 78924 33264
1 75 2130 29790 224193 931275 2092220 2322060 964656
This here is the first subdiagonal. The diagonal is A135407. The 2nd column is A001610, the 3rd A242300, the 4th A213807. (End)

Cf. A203010.

f[k_] := LucasL[k - 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203009 *)

A254232 Product of Perrin numbers A001608(2) * ... * A001608(n).

Original entry on oeis.org

2, 6, 12, 60, 300, 2100, 21000, 252000, 4284000, 94248000, 2733192000, 106594488000, 5436318888000, 369669684384000, 33270271594560000, 3959162319752640000, 625547646520917120000, 130739458122871678080000, 36214829900035454828160000

Offset: 2

Vaclav Kotesovec, Jan 27 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..127

Cf. A001608, A003266, A060006, A126772, A135407, A254231.

Table[Product[SeriesCoefficient[(3-x^2)/(1-x^2-x^3),{x,0,k}],{k,2,n}], {n,2,20}]

A254232_list, a, b, c, d = [2], 3, 0, 2, 2
for _ in range(200):
    a, b, c = b, c, a+b
    d *= c
    A254232_list.append(d) # Chai Wah Wu, Jan 28 2015

a(n) ~ c * r^(n*(n+1)/2), where r = A060006 = 1.324717957244746025960908854478... is the root of the equation r^3 = r + 1, c = 0.81845731383668335747954234022593868885066763327809025622515304041339344876... .

A294349 Product of first n terms of the binomial transform of the Lucas numbers (A000032).

Original entry on oeis.org

2, 6, 42, 756, 35532, 4370436, 1407280392, 1186337370456, 2618246576596392, 15128228719573952976, 228844715840995186667952, 9062937281450932377610903056, 939663463215395570775453650652192, 255065069445576619918001465293982953056

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A005248, A047656, A086619, A135407, A194157.

Table[Product[Sum[Binomial[m, k]*LucasL[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]
Table[Product[LucasL[2*k], {k, 0, n}], {n, 0, 15}]

a(n) ~ c * phi^(n*(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 2.349094356918735309421297337651771419003525539652230102934874983942...

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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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A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

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A218490 Decimal expansion of Lucas factorial constant.

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A256832 Product of first n Pell numbers Pell(1), ... , Pell(n).

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

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A254231 Product of tribonacci numbers A000073(2) * ... * A000073(n).

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A203009 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(0)=2.

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