A176343 -id:A176343 - cp's OEIS frontend

A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 11, 13, 11, 1, 1, 65, 75, 75, 65, 1, 1, 568, 632, 640, 632, 568, 1, 1, 7789, 8356, 8418, 8418, 8356, 7789, 1, 1, 168761, 176549, 177114, 177168, 177114, 176549, 168761, 1, 1, 5847568, 6016328, 6024114, 6024671, 6024671, 6024114, 6016328, 5847568, 1

Offset: 0

Roger L. Bagula, Apr 15 2010

			Triangle begins:
  1;
  1,      1;
  1,      1,      1;
  1,      3,      3,      1;
  1,     11,     13,     11,      1;
  1,     65,     75,     75,     65,      1;
  1,    568,    632,    640,    632,    568,      1;
  1,   7789,   8356,   8418,   8418,   8356,   7789,      1;
  1, 168761, 176549, 177114, 177168, 177114, 176549, 168761, 1;
  ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A156070, A156072, A176305, A176306, A176307, A176625, A176339.

b:= function(n)
    if n=0 then return 0;
    else return 1 + Fibonacci(n)*b(n-1);
    fi; end;
T:= function(n,k) return 1 + b(n) - b(n-k) - b(k); end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 07 2019

function b(n)
  if n eq 0 then return 0;
  else return 1 + Fibonacci(n)*b(n-1);
  end if; return b; end function;
function T(n,k) return 1 + b(n) - b(n-k) - b(k); end function; [ T(n,k) : k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019

with(combinat);
b:= proc(n) option remember;
   if n = 0 then 0    else 1+fibonacci(n)*b(n-1)
   fi; end proc;
T:= proc (n, k) 1+b(n)-b(n-k)-b(k) end proc;
seq(seq(T(n, k), k = 0..n), n = 0..10); # G. C. Greubel, Dec 08 2019

b[n_]:= b[n]= If[n==0, 0, Fibonacci[n]*b[n-1] + 1]; (* A176343 *)
T[n_, k_]:= T[n, k] = 1 + a[n] - a[n-k] - a[k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Dec 08 2019 *)

(a[0] : 0, a[n] := fib(n)*a[n-1] + 1, T(n, m) := 1 + a[n] - a[m] - a[n-m])$ create_list(T(n, m), n, 0, 10, m, 0, n); /* Franck Maminirina Ramaharo, Nov 25 2018 */

b(n) = if(n==0, 0, 1 + fibonacci(n)*b(n-1) );
T(n,k) = 1 + b(n) - b(n-k) - b(k);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 07 2019

def b(n):
    if (n==0): return 0
    else: return 1 + fibonacci(n)*b(n-1)
def T(n,k): return 1 + b(n) - b(n-k) - b(k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019

Edited and name clarified by Franck Maminirina Ramaharo, Nov 25 2018

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

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).

cp's OEIS Frontend

A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).

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

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

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

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