A101689 -id:A101689 - cp's OEIS frontend

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

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

A176343 a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

Original entry on oeis.org

0, 1, 2, 5, 16, 81, 649, 8438, 177199, 6024767, 331362186, 29491234555, 4246737775921, 989489901789594, 373037692974676939, 227552992714552932791, 224594803809263744664718, 358677901683394200229554647, 926823697949890613393169207849

Offset: 0

Roger L. Bagula, Apr 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000045, A003266, A062073, A101689, A139339, A238243, A238244.

a:= function(n)
    if n=0 then return 0;
    else return 1 + Fibonacci(n)*a(n-1);
    fi; end;
List([0..20], n-> a(n) ); # G. C. Greubel, Dec 07 2019

function a(n)
  if n eq 0 then return 0;
  else return 1 + Fibonacci(n)*a(n-1);
  end if; return a; end function;
[a(n): n in [0..20]]; // G. C. Greubel, Dec 07 2019

with(combinat);
a:= proc(n) option remember;
      if n=0 then 0
    else 1 + fibonacci(n)*a(n-1)
      fi; end:
seq( a(n), n=0..20); # G. C. Greubel, Dec 07 2019

a[n_]:= a[n]= If[n==0, 0, Fibonacci[n]*a[n-1] +1]; Table[a[n], {n,0,20}]

a(n) = if(n==0, 0, 1 + fibonacci(n)*a(n-1) ); \\ G. C. Greubel, Dec 07 2019

def a(n):
    if (n==0): return 0
    else: return 1 + fibonacci(n)*a(n-1)
[a(n) for n in (0..20)] # G. C. Greubel, Dec 07 2019

a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * A101689 = 3.317727324507285486862890025085971028467... is product of Fibonacci factorial constant (see A062073) and Sum_{n>=1} 1/(Product_{k=1..n} A000045(k) ). - Vaclav Kotesovec, Feb 20 2014

A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

Original entry on oeis.org

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

Offset: 1

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

			2.8702221569733963308194588658111996012403192622809957012...

Engel, F. "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A006784, A064648, A101689, A001405, A070910, A096789, A141827.

evalf(BesselI(0, 2) + BesselI(1, 2) - 1, 100); # Peter Bala, Jul 02 2016

First@ RealDigits@ N[Sum[1/Product[Ceiling[r/2], {r, n}], {n, 1000}], 100] (* Original program amended to generate output by Michael De Vlieger, Jul 03 2016 *)
RealDigits[3 - HypergeometricPFQ[{1, 1}, {3, 3, 3}, 1]/8, 10, 100][[1]] (* Vaclav Kotesovec, Jul 03 2016 *)

From Peter Bala, Jul 01 2016: (Start)
Constant c = 1/1 + 1/(1*1) + 1/(1*1*2) + 1/(1*1*2*2) + 1/(1*1*2*2*3) + 1/(1*1*2*2*3*3) + ... = Sum_{n >= 1} binomial(n,floor(n/2))/n!.
Alternative series representations:
c = 3 - Sum_{n >= 2} 1/(n*(n - 1)*n!^2);
c = 1 + Sum_{n >= 1} (n + 2)/(n!*(n + 1)!);
c = 5/3 + 1/3*Sum_{n >= 2} (n + 1)*(n + 2)/n!^2;
c = A070910 + A096789 - 1.
Continued fraction: c = 3 - 1/(8 - 4/(14 - 9/(32 - ... - (n-1)^2/(n^2 + n + 2 - ...)))). See comments in A141827. (End)

A238244 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 3.

Original entry on oeis.org

1, 4, 11, 36, 183, 1467, 19074, 400557, 13618941, 749041758, 66664716465, 9599719170963, 2236734566834382, 843248931696562017, 514381848334902830373, 507694884306549093578154, 810788730237558902444311941, 2095078078933852203916102055547

Offset: 1

Vaclav Kotesovec, Feb 20 2014

nonn

Generally, sequence a(n) = Fibonacci(n)*a(n-1) + p, with a(1)=1 and fixed p, is asymptotic to c(p) * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where constant c(p) = A062073 * (p*A101689 - p + 1).

Harvey P. Dale, Table of n, a(n) for n = 1..98

Cf. A176343, A238243, A003266, A101689, A062073, A000045, A139339.

RecurrenceTable[{a[n]==Fibonacci[n]*a[n-1]+3,a[1]==1},a,{n,1,20}]
nxt[{n_,a_}]:={n+1,a*Fibonacci[n+1]+3}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 04 2024 *)

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * (3*A101689-2) = 7.4996979520811499717534... is product of Fibonacci factorial constant (see A062073) and -2+3*sum_{n>=1} 1/product(A000045(k), k=1..n).

A137991 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of Fibonacci numbers (A000045) as coefficients.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Pierce Expansion.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A000045, A101689 (reciprocal), A137987.

with (combinat,fibonacci); P:=proc(n) local a,i,k; a:=0; k:=1; for i from 1 by 1 to n do k:=k*fibonacci(i); a:=a+1/k; print(evalf(1/a,100)); od; end: P(100);

RealDigits[N[1/(Sum[Product[1/Fibonacci[k], {k, 1, n}], {n, 1, 1000}]),
100]][[1]] (* G. C. Greubel, Dec 26 2016 *)

A238243 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 2.

Original entry on oeis.org

1, 3, 8, 26, 132, 1058, 13756, 288878, 9821854, 540201972, 48077975510, 6923228473442, 1613112234311988, 608143312335619478, 370967420524727881582, 366144844057906419121436, 584733315960476551336933294, 1510950888441871408654635631698

Offset: 1

Vaclav Kotesovec, Feb 20 2014

nonn

Cf. A176343, A238244, A003266, A101689, A062073, A000045, A139339.

RecurrenceTable[{a[n]==Fibonacci[n]*a[n-1]+2,a[1]==1},a,{n,1,20}]

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * (2*A101689-1) = 5.4087126382942177293... is product of Fibonacci factorial constant (see A062073) and -1+2*sum_{n>=1} 1/product(A000045(k), k=1..n).

A101690 Decimal expansion of the unique real number x whose Engel expansion is the Lucas sequence.

Original entry on oeis.org

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

Offset: 1

Ryan Propper, Dec 11 2004

			x = 1.4297159226892042002772306926271655374969467995845816636429774710...

Cf. A101689, A000204.

Lucas[n_Integer?Positive] := Lucas[n] = Lucas[n-1] + Lucas[n-2]; Lucas[1] = 1; Lucas[2] = 3; N[Sum[1/Product[Lucas[i], {i, n}], {n, 500}], 100]
digits = 100; Clear[x]; x[m_] := x[m] = N[Sum[1/Product[LucasL[i], {i, 1, n}], {n, 1, m}], digits+5]; m = 10; While[x[m] != x[m-1], m++]; RealDigits[x[m], 10, digits][[1]] (* Jean-François Alcover, Nov 20 2015 *)

x = Sum_{n >= 1} 1/(Product_{1 <= i <= n} L(i)), where L(i) is the i-th Lucas number.

Offset corrected by Amiram Eldar, Nov 09 2020

A130818 Decimal expansion of number whose Engel expansion is the sequence of squares, that is, 1, 4, 9, 16,...

Original entry on oeis.org

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

Offset: 1

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

			1.2795853023360672674372044408115333532858411...

F. Engel "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012, page 17.
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind

Cf. A006784, A064648, A101689.

RealDigits[BesselI[0, 2] - 1, 10, 105] // First (* Jean-François Alcover, Oct 01 2013 *)

besseli(0,2)-1 \\ Charles R Greathouse IV, Oct 01 2013

Equal to Sum_{n>=1} 1/n!^2 or BesselI(0,2) - 1. - Gerald McGarvey, Nov 12 2007

Equals A070910 - 1. - R. J. Mathar, Jun 13 2008

cp's OEIS Frontend

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

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A176343 a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

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A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

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A238244 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 3.

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A137991 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of Fibonacci numbers (A000045) as coefficients.

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A238243 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 2.

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A101690 Decimal expansion of the unique real number x whose Engel expansion is the Lucas sequence.

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