A126772 -id:A126772 - cp's OEIS frontend

A172353 Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 4, 2, 1, 1, 3, 6, 12, 6, 3, 1, 1, 4, 12, 24, 24, 12, 4, 1, 1, 5, 20, 60, 60, 60, 20, 5, 1, 1, 7, 35, 140, 210, 210, 140, 35, 7, 1, 1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1

Offset: 0

Roger L. Bagula, Feb 01 2010

Start from the Padovan sequence A134816 and its partial products A126772, extended by A126772(0)=1. Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 4, 8, 14, 32, 82, 232, 786, 2981,..

			1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 4, 2, 1;
1, 3, 6, 12, 6, 3, 1;
1, 4, 12, 24, 24, 12, 4, 1;
1, 5, 20, 60, 60, 60, 20, 5, 1;
1, 7, 35, 140, 210, 210, 140, 35, 7, 1;
1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1;

Cf. A010048, A172355

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

Original entry on oeis.org

1, 1, 1, 2, 6, 30, 240, 3120, 65520, 2227680, 122522400, 10904493600, 1570247078400, 365867569267200, 137932073613734400, 84138564904377984000, 83044763560621070208000, 132622487406311849122176000, 342696507457909818131702784000

Offset: 0

N. J. A. Sloane

Equals right border of unsigned triangle A158472. - Gary W. Adamson, Mar 20 2009
Three closely related sequences are A194157 (product of first n nonzero F(2*n)), A194158 (product of first n nonzero F(2*n-1)) and A123029 (a(2*n) = A194157(n) and a(2*n-1) = A194158(n)). - Johannes W. Meijer, Aug 21 2011
a(n+1)^2 is the number of ways to tile this pyramid of height n with squares and dominoes, where vertical dominoes can only appear (if at all) in the central column. Here is a pyramid of height n=4,
       _
     ||_
   ||_||
 ||_|||_|_
|||_|||_|_|,
and here is one of the a(5)^2 = 900 possible such tilings with our given restrictions:
       _
     | |
   ||_||
 |__|_|_|_
||__|___|||. - Greg Dresden and Jiayi Liu, Aug 23 2024

			a(5) = 30 because the first 5 Fibonacci numbers are 1, 1, 2, 3, 5 and 1 * 1 * 2 * 3 * 5 = 30.
a(6) = 240 because 8 is the sixth Fibonacci number and a(5) * 8 = 240.
a(7) = 3120 because 13 is the seventh Fibonacci number and a(6) * 13 = 3120.
G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 30*x^5 + 240*x^6 + 3120*x^7 + ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison Wesley, p 597
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..99 (terms n = 1..50 from T. D. Noe)
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda, Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C^2-finite Sequences, in Applied Mathematical Analysis and Computations (SGMC 2021) Springer Proc. Math. Stat., Vol. 471. Springer, Cham, 255-297. See p. 287.
Mathematica Stack Exchange, Product of Fibonacci numbers using For/Do/While loops.
Yuri V. Matiyasevich and Richard K. Guy, A new formula for Pi, Amer. Math. Monthly 93 (1986), no. 8, 631-635. Math. Rev. 2000i:11199.
Aidan Sudbury, Arthur Sun, David Treeby, and Edward Wang, Pick-up Sticks and the Fibonacci Factorial, arXiv:2504.19911 [math.PR], 2025.
Thotsaporn Aek Thanatipanonda and Yi Zhang, Sequences: Polynomial, C-finite, Holonomic, ..., arXiv:2004.01370 [math.CO], 2020. See pp. 5-6.
Eric Weisstein's World of Mathematics, Fibonorial
Index to divisibility sequences.

Cf. A000045, A101689, A135598, A158472.
Cf. A123741 (for Fibonacci second version), A002110 (for primes), A070825 (for Lucas), A003046 (for Catalan), A126772 (for Padovan), A069777 (q-factorial numbers for sums of powers). - Johannes W. Meijer, Aug 21 2011
Cf. A176343, A238243, A238244.

a003266 n = a003266_list !! (n-1)
a003266_list = scanl1 (*) $ tail a000045_list
-- Reinhard Zumkeller, Sep 03 2013

with(combinat): A003266 := n-> mul(fibonacci(i),i=1..n): seq(A003266(n), n=0..20);

Rest[FoldList[Times,1,Fibonacci[Range[20]]]] (* Harvey P. Dale, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, Fibonorial[n]]; (* Michael Somos, Oct 23 2017 *)
Table[Round[GoldenRatio^(n(n-1)/2) QFactorial[n, GoldenRatio-2]], {n, 20}] (* Vladimir Reshetnikov, Sep 14 2016 *)

a(n)=prod(i=1,n,fibonacci(i)) \\ Charles R Greathouse IV, Jan 13 2012

from itertools import islice
def A003266_gen(): # generator of terms
    a,b,c = 1,1,1
    while True:
        yield c
        c *= a
        a, b = b, a+b
A003266_list = list(islice(A003266_gen(),20)) # Chai Wah Wu, Jan 11 2023

a(n) is asymptotic to C*phi^(n*(n+1)/2)/sqrt(5)^n where phi = (1 + sqrt(5))/2 is the golden ratio and the decimal expansion of C is given in A062073. - Benoit Cloitre, Jan 11 2003
a(n+3) = a(n+1)*a(n+2)/a(n) + a(n+2)^2/a(n+1). - Robert Israel, May 19 2014
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 06 2014
0 = a(n)*(+a(n+1)*a(n+3) - a(n+2)^2) + a(n+2)*(-a(n+1)^2) for all n >= 0. - Michael Somos, Oct 06 2014
Sum_{n>=1} 1/a(n) = A101689. - Amiram Eldar, Oct 27 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = A135598. - Amiram Eldar, Apr 12 2021
a(n) = (2/sqrt(5))^n * Product_{j=1..n} i^j*sinh(c*j), where c = arccsch(2) - i*Pi/2. - Peter Luschny, Jul 07 2025

a(0)=1 prepended by Alois P. Heinz, Oct 12 2016

A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Mar 14 2001

Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number.
The name "plastic number" goes back to the Dutch Benedictine monk and architect Dom Hans van der Laan, who gave this name 4 years after the discovery of the number by the French engineer Gérard Cordonnier in 1924, who used the name "radiant number". - Hugo Pfoertner, Oct 07 2018
Sometimes denoted by the symbol rho. - Ed Pegg Jr, Feb 01 2019
Also the solution of 1/x + 1/(1+x+x^2) = 1. - Clark Kimberling, Jan 02 2020
Given any complex p such that real(p)>-1, this constant is the only real solution of the equation z^p+z^(p+1)=z^(p+3), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+3)), convergent from any complex plane point. - Stanislav Sykora, Oct 14 2021
The Pisot-Vijayaraghavan numbers were named after the French mathematician Charles Pisot (1910-1984) and the Indian mathematician Tirukkannapuram Vijayaraghavan (1902-1955). - Amiram Eldar, Apr 02 2022
The sequence a(n) = v_3^floor(n^2/4) where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1 satisfies the Somos-5 recursion a(n+3)*a(n-2) = a(n+2)*a(n-1) + a(n+1)*a(n) for all n in Z. Also true if floor is removed. - Michael Somos, Mar 24 2023

			1.32471795724474602596090885447809734...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Alex Bellos, The golden ratio has spawned a beautiful new curve: the Harriss spiral, The Guardian, Jan 13 2015.
Gamaliel Cerda-Morales, New Identities for Padovan Numbers, arXiv:1904.05492 [math.CO], 2019.
Brady Haran and Edmund Harriss, The Plastic Ratio, Numberphile video (2019).
Ed Pegg Jr., Pictures based on the plastic constant
Simon Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits
Simon Plouffe, The Smallest Pisot-Vijayaraghavan number
F. Rothelius, Formulae.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden, 1960.
Michel Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Maverick Graph.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Plastic Constant.
Wikipedia, Plastic number.
Index entries for algebraic numbers, degree 3

Cf. A001622. A072117 gives continued fraction.
Cf. A006888, A010527, A051016, A051017, A084252, A075778 (inverse), A126772.
Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.
Cf. A002620, A006721, A134816.

SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // G. C. Greubel, Mar 15 2019

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%,130) ; # R. J. Mathar, Jan 22 2013

RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Sep 30 2009 *)
s = Sqrt[23/108]; RealDigits[(1/2 + s)^(1/3) + (1/2 - s)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 12 2017 *)
RealDigits[Root[x^3-x-1,1],10,120][[1]] (* or *) RealDigits[(Surd[9-Sqrt[69],3]+Surd[9+Sqrt[69],3])/(Surd[2,3]Surd[9,3]),10,120][[1]] (* Harvey P. Dale, Sep 04 2018 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060006.txt", n, " ", d)); \\ Harry J. Smith, Jul 01 2009

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ Altug Alkan, Apr 10 2016

polrootsreal(x^3-x-1)[1] \\ Charles R Greathouse IV, Aug 28 2016

default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* Michael Somos, Mar 24 2023 */

numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # G. C. Greubel, Mar 15 2019

Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003
Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004
Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006
Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008
Equals Sum_{k >= 0} 27^(-k)/k!*(Gamma(2*k+1/3)/(9*Gamma(k+4/3)) - Gamma(2*k-1/3)/(3*Gamma(k+2/3))). - Robert Israel, Jan 13 2015
Equals sqrt(Phi) = sqrt(1.754877666246...) (see A109134). - Philippe Deléham, Sep 29 2020
Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - Amiram Eldar, May 15 2021
Equals 1/hypergeom([1/5, 2/5, 3/5, 4/5], [2/4, 3/4, 5/4], -5^5/4^4). - Gerry Martens, Mar 16 2025

Edited and extended by Robert G. Wilson v, Aug 03 2002

Removed incorrect comments, Joerg Arndt, Apr 10 2016

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

A253924 Decimal expansion of Padovan factorial constant.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 26 2015

The Padovan factorial constant is associated with the Padovan factorial A126772.

			1.25373683131537208838997864311903035079685338006712312402418098138...

Eric Weisstein's World of Mathematics, Padovan Sequence

Cf. A000931, A060006, A126772.

Equals limit n->infinity A126772(n) / (d^(n/2) * r^(n^2/2)), where r = 1.324717957244746025960908854478... (see A060006) is the root of the equation r^3 = r + 1, d = 0.3936412824011163853866584484465616545579227324... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0.

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... .

cp's OEIS Frontend

A172353 Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).

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A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

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A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

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

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A253924 Decimal expansion of Padovan factorial constant.

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

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

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