A062073 -id:A062073 - cp's OEIS frontend

A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

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

Offset: 0

Johannes W. Meijer, Aug 21 2011

a(n) = Product_{i=1..n} F(2*i) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi = (1+sqrt(5))/2 and F(n) = A000045(n), see A194157. The decimal expansion of the constant C2 is given above.

			C2 = 0.83288324403391298245025664...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

Eric Weisstein, Fibonorial Mathworld.

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

Cf. A349272.

digits = 108; NProduct[1 - GoldenRatio^(-4*k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013, from 1st formula *)
RealDigits[QPochhammer[1/GoldenRatio^4], 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

Equals Product_{k>=1} (1-alpha^(2*k)) with alpha = -1/phi^2 and phi = (1+sqrt(5))/2.
Equals Sum_{n>=0} (-1)^binomial(n+1,2)*alpha^A152749(n).
Equals A062073/A194160.

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

A276987 Decimal expansion of (phi-1)_inf = (1/phi)_inf, where (q)_inf is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Sep 24 2016

(1/phi)_inf appears as a coefficient in asymptotics of A274983, A274985.

			0.1208019218617061294237231569887920563...

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Golden Ratio.

Cf. A274983, A274985, A062073, A227681.

RealDigits[QPochhammer[1/GoldenRatio], 10, 100][[1]]

(1/phi)inf = Product{k > 0} (1 - 1/phi^k).

A003268 Central Fibonomial coefficients.

Original entry on oeis.org

1, 2, 6, 15, 60, 260, 1820, 12376, 136136, 1514513, 27261234, 488605194, 14169550626, 411591708660, 19344810307020, 908637119420910, 69056421075989160, 5249543573067466872, 645693859487298425256, 79413089729752455762384, 15803204856220738696714416

Offset: 0

N. J. A. Sloane

nonn

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 74.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

Central column of A010048, |A055870|.

Cf. A062073.

Table[Product[Fibonacci[k],{k,Floor[n/2]+1,n}] / Product[Fibonacci[k],{k,1,Ceiling[n/2]}],{n,2,20}] (* Vaclav Kotesovec, Apr 10 2015 *)

a(n) = (Product_{k=floor(n/2)+1..n} Fibonacci(k)) / (Product_{k=1..ceiling(n/2)} Fibonacci(k)).

a(n) ~ ((1+sqrt(5))/2)^(n^2/4 + n + 1 - (1-(-1)^n)/8) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant. - Vaclav Kotesovec, May 01 2015

More terms from Vaclav Kotesovec, May 01 2015

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

A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Aug 21 2011

A194158(n) = prod(i=1..n, F(2*i-1) ) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 and F(n) = A000045(n). The decimal expansion of the constant C1 is given here.

			C1 = 1.4728859290995693146071...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

Eric Weisstein, Fibonorial, Mathworld.

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

RealDigits[Product[1-((-1)/GoldenRatio^2)^(2k-1),{k,1000}],10, 100] [[1]] (* Harvey P. Dale, Aug 28 2011 *)
RealDigits[QPochhammer[-GoldenRatio^2, 1/GoldenRatio^4]/(GoldenRatio Sqrt[5]), 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

C1 = prod(k>=1, 1-alpha^(2*k-1) ) where alpha = (-1/phi^2) and phi = (1+sqrt(5))/2.

C1 = A062073/A194159

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)).

A181926 Diagonal sums of Fibonomial triangle A010048.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 13, 27, 70, 191, 609, 2130, 8526, 38156, 194000, 1109673, 7176149, 52238676, 429004471, 3970438003, 41454181730, 488046132076, 6482590679282, 97134793638750, 1641654359781521, 31285014253070731, 672372121341768918, 16299021330860540657

Offset: 0

Emanuele Munarini, Apr 02 2012

Cf. A000045 (Fibonacci) as diagonal sums of A007318 (Pascal's Triangle). For Fibonacci numbers, the ratio A000045(i+1)/A000045(i) approaches the golden ratio (1+sqrt(5))/2 as i increases. For this sequence, it appears that (a(i+5)/a(i+4))/(a(i+1)/a(i)) approaches the golden ratio. - Dale Gerdemann, Apr 23 2015

This sequence can be interpreted as counting colored, square-domino tilings of a 1xn board, where the number of colors available for a domino with k squares to the left is Fib(k+1) and the number of colors available for a square with k dominoes to the left is Fib(k-1). "Fib(n)" here is A000045 (Fibonacci), extended so that Fib(-1) = 1, Fib(0) = 0,... . As an example, let d be a domino, s be a square an consider the uncolored tilings of length 5: sssss, sssd, ssds, sdss, dsss, sdd, dsd, dds. Then, after each 's' or 'd', write the number of colors available: s1s1s1s1s1, s1s1s1d3, s1s1d2s0, s1d1s0s0, d1s0s0s0, s1d1d1, d1s0d1, d1d1s1. So the number of colorings for these tilings is: 1,3,0,0,0,1,0,1 and the total number of colored tilings is 6 (= a(5)). - Dale Gerdemann, Apr 30 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..195
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A003266, A010048, A056569, A062073.

Table[Sum[Product[Fibonacci[k-j+1]/Fibonacci[j],{j,1,n-k}],{k,Ceiling[n/2],n}],{n,0,30}] (* Vaclav Kotesovec, Apr 29 2015 *)
(* Or, since version 10 *) Table[Sum[Fibonorial[k]/Fibonorial[2k-n]/Fibonorial[n-k],{k,Ceiling[n/2],n}],{n,0,30}] (* Vaclav Kotesovec, Apr 30 2015 *)
(* List of the multiplicative constants from an asymptotic formula: *) {N[EllipticTheta[3, 0, GoldenRatio^(-2)]/QPochhammer[-(GoldenRatio^2)^(-1)], 80], N[Sum[GoldenRatio^(-2*(j + 1/4)^2), {j, -Infinity, Infinity}]/QPochhammer[-(GoldenRatio^2)^(-1)], 80], N[EllipticTheta[2, 0, GoldenRatio^(-2)]/QPochhammer[-(GoldenRatio^2)^(-1)], 80]} (* Vaclav Kotesovec, Apr 30 2015 *)

ffib(n):=prod(fib(k),k,1,n);
fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));
makelist(sum(fibonomial(k,n-k),k,ceiling(n/2),n),n,0,30);

a(n) = sum(fibonomial(k,n-k),k=ceiling(n/2)..n).
From Vaclav Kotesovec, Apr 29 2015: (Start)
a(n) ~ c * ((1+sqrt(5))/2)^(n^2/8), where
c = 1.472885929099569314607134281503815932269629515265... if mod(n,4)=0,
c = 1.472782295338429619549807628338486893461428897618... if mod(n,4)=1 or 3,
c = 1.472678661577289942545896597162143392952724631588... if mod(n,4)=2.
Or c = Sum_{j} ((1+sqrt(5))/2)^(-2*(j+(1-cos(Pi*n/2))/4)^2) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant.
(End)
a(n) = Sum_{k=ceiling(n/2)..n} A003266(k) / (A003266(2*k-n) * A003266(n-k)). - Vaclav Kotesovec, Apr 30 2015

a(14) corrected by Vaclav Kotesovec, Apr 29 2015

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).

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.

cp's OEIS Frontend

A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

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A218490 Decimal expansion of Lucas factorial constant.

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A276987 Decimal expansion of (phi-1)_inf = (1/phi)_inf, where (q)_inf is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

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A003268 Central Fibonomial coefficients.

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

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A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

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A256831 Decimal expansion of Pell factorial constant.

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