A062073 -id:A062073 - cp's OEIS frontend

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

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).

A253267 Decimal expansion of a constant related to A152686.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 01 2015

			1.0964140725073244231102159988444403759459296087776979384650568137801405219...

Cf. A062073, A062381, A152686.

(* Iterations: *) Do[Print[N[Product[Product[Fibonacci[k],{k,1,j}],{j,1,n}] / (GoldenRatio^(n*(n+1)*(n+2)/6) * QPochhammer[-1/GoldenRatio^2]^n / 5^(n*(n+1)/4)),120]],{n,100,1000,100}]

Equals limit n->infinity A152686(n) / (((1+sqrt(5))/2)^(n*(n+1)*(n+2)/6) * A062073^n / 5^(n*(n+1)/4)).

A062072 Continued fraction expansion of Fibonacci factorial constant.

Original entry on oeis.org

1, 4, 2, 2, 3, 2, 15, 9, 1, 2, 1, 2, 15, 7, 6, 21, 3, 5, 1, 23, 1, 11, 1, 7, 1, 3, 1, 12, 2, 1, 1, 1, 7, 1, 3, 1, 12, 2, 1, 2, 2, 9, 27, 1, 1, 1, 1, 2, 19, 3, 8, 1, 1, 15, 3, 1, 2, 1, 1, 1, 3, 2, 3, 8, 1, 1, 14, 1, 49, 2, 1, 17, 4, 2, 1, 2, 2, 1, 3, 1, 5, 1, 1, 3, 1, 2, 1, 4, 1, 2, 5, 1, 3, 2, 1, 1, 2, 6

Offset: 0

Jason Earls, Jun 27 2001

			1.2267420107203532444176302...

R. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.

Harry J. Smith, Table of n, a(n) for n = 0..4999
Simon Plouffe, Plouffe's Inverter

Cf. A062073 (decimal expansion).

\p 500 a=-1/(1/2+sqrt(5)/2)^2; contfrac(prod(n=1,17000,(1-a^n)))

{ allocatemem(932245000); default(realprecision, 5300); p=-1/(1/2 + sqrt(5)/2)^2; x=contfrac(prodinf(k=1, 1-p^k)); for (n=1, 5000, write("b062072.txt", n-1, " ", x[n])) } \\ 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.

Offset changed by Andrew Howroyd, Aug 04 2024

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset: 1

Vaclav Kotesovec, May 05 2012

nonn

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
V. Kotesovec, Non-attacking chess pieces

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

A191994 (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).

Original entry on oeis.org

1, 2, 8, 42, 360, 4800, 102960, 3538080, 196035840, 17520703200, 2529842515200, 590412901478400, 222813349683724800, 136001024583142118400, 134285149587387262464000, 214504624277084224347264000, 554361997358383529330695680000

Offset: 1

K.V.Iyer, Venkata Subba Reddy P., Charles R Greathouse IV, Jun 21 2011

Let F(1), F(2), F(3), ... be the Fibonacci numbers 1, 1, 2, .... For k=1, we define the tree T(1) the path on two vertices with one identified as the root r. We assign the edge-weight F(1). T(2) is obtained from T(1) by attaching F(2) vertex to the pendents in T(1) except r. In T(2), r is retained as in T(1) and the new edge-weight is assigned as F(2). For k>1, T(k) is obtained from T(k-1) by attaching F(k) vertices to pendents in T(k-1) except r. In T(k), r is retained as in T(k-1) and all the new edge-weights are assigned F(k). With D(1)=1, for k>1 let D(k)=Sum of all distances d(r,x) taken across all vertices x in T(k). By induction it follows that for k>1, D(k)-D(k-1) is this sequence.

Retaining the notation of D(k) above, it follows, for k>1, that if D(k)=a(1)F(1)+ - - - +a(k)F(k) then D(k+1)=b(1)F(1)+ - - - +b(k)F(k)+b(k+1)F(k+1) where b(k+1) is the number of leaf nodes in T(k+1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..97
Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Cf. A000071 (sum of Fibonacci numbers), A003266 (product of Fibonacci numbers).

Cf. A062073 (Fibonacci factorial constant).

s=0;p=1;for(n=1,40,f=fibonacci(n);s+=f;p*=f;print1(s*p", ")) \\ Charles R Greathouse IV, Jun 21 2011

a(n)=prod(k=1,n,fibonacci(k))*(fibonacci(n+2)-1) /* Franklin T. Adams-Watters, Jun 23 2011 */

a(n) ~ C*sqrt(phi^(n^2 + 3*n + 4)/5^(n+1)) where C = A062073 and phi = (1+sqrt(5))/2.

a(n) = (F(n+2)-1) * Product_{k=1..n} F(k). - Franklin T. Adams-Watters, Jun 23 2011

A382910 a(n) = A003266(n)^2.

Original entry on oeis.org

1, 1, 1, 4, 36, 900, 57600, 9734400, 4292870400, 4962558182400, 15011738501760000, 118907980672440960000, 2465675887223735746560000, 133859078241489389944995840000, 19025256931384645503492313743360000, 7079298104168226591849489943904256000000, 6896432754839457130755425769163265163264000000

Offset: 0

Edwin Hermann, Apr 08 2025

For n>=3 number of valid symmetrical change ringing methods on n bells with the shortest number of rows per lead where the treble plain hunts out to the back. See Wikipedia and the Polster and Ross link for an explanation of bell ringing terminology.

Alois P. Heinz, Table of n, a(n) for n = 0..70
Richard Duckworth and Fabian Stedman, Tintinnalogia, or, the Art of Ringing, (1671). Released by Project Gutenberg, 2006.
Burkard Polster and Marty Ross, Ringing the changes, (2009).
Wikipedia, Method ringing.
Index entries for sequences related to bell ringing.

Cf. A000045, A001622, A003266, A007598, A062073, A090281.

a:= proc(n) a(n):= `if`(n=0, 1, a(n-1)*(<<0|1>, <1|1>>^n)[1, 2]^2) end:
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 14 2025

k = 1; {1, 1}~Join~Reap[Do[k *= Fibonacci[n]; Sow[k^2], {n, 16}] ][[-1, 1]] (* Michael De Vlieger, Apr 14 2025 *)

a(n) = Product_{j=1..n} Fibonacci(j)^2.
a(0) = 1; a(n) = a(n-1)*A007598(n). - Hugo Pfoertner, Apr 13 2025
a(n) ~ c^2 * phi^(n*(n+1)) / 5^n where phi is the golden ratio  (A001622) and c = A062073. - Amiram Eldar, Aug 18 2025

A094221 1/detM(n) where M(n) is the n X n matrix m(i,j)=F(i)/F(i+j-1) and F(i)=i-th Fibonacci number.

Original entry on oeis.org

1, -2, -180, 2808000, 63248290560000, -13040516214928232110080000, -173699422048124050990739961787485511680000, 1013027110717881203216509560866301885575342298295136595148800000

Offset: 1

Benoit Cloitre, May 28 2004

sign

Cf. A062381.

Table[(-1)^Floor[n/2] * Product[Fibonacci[k]^k,{k,1,n-1}] * Product[Fibonacci[k]^(2*n-k),{k,n,2*n-1}] / Product[Fibonacci[k],{k,1,n}] / Product[Product[Fibonacci[k],{k,1,j-1}],{j,1,n}]^2,{n,1,10}] (* Vaclav Kotesovec, May 01 2015 *)

a(n)=1/matdet(matrix(n,n,i,j,fibonacci(i)/(fibonacci(i+j-1))))

a(n) = A062381(n)/A003266(n). - corrected by Vaclav Kotesovec, May 01 2015

a(n) ~ (-1)^floor(n/2) * A253270 * ((1+sqrt(5))/2)^(n*(4*n^2 - 3*n - 1)/6) / (A253267^2 * A062073^(2*n-1)). - Vaclav Kotesovec, May 01 2015

A241990 Decimal expansion of 'delta', a constant arising in the asymptotics of the regularized product of the Fibonacci numbers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 11 2014

			0.899212680785500886257698838775288182435045411706848498172656...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5 Fibonacci factorials, p. 10.

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 1.
Adrian R. Kitson, The regularized product of the Fibonacci numbers. (2006) arXiv:math/0608187 [math.HO]

Cf. A062073.

c = QPochhammer[-1/GoldenRatio^2]; delta = 5^(1/4)*Exp[-Log[5]^2/(8*Log[GoldenRatio])]*c/GoldenRatio^(1/12); RealDigits[delta, 10, 103] // First

delta = 5^(1/4)*exp(-log(5)^2/(8*log(phi)))*c/phi^(1/12), where phi is the golden ratio and c is the Fibonacci factorial constant (c = A062073 = 1.226742...).

A271421 a(n) = fibonorial(3n)/(fibonorial(2n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).

Original entry on oeis.org

1, 4, 119, 23496, 32149806, 300214157831, 19246160432331107, 8451529006578585976752, 25443734373070679510011112460, 524973397889459587964008354031908560, 74243674067972394056586805754940632245000310, 71965837912588688126721254257169744333502564695515911

Offset: 1

Vladimir Reshetnikov, May 21 2016

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.

Simon Plouffe, Fibonacci factorials.
Eric Weisstein's World of Mathematics, Fibonorial, Fibonacci Factorial Constant.

Cf. A003266, A003267, A003268, A062073, A003150, A000045.

Table[Fibonorial[3 n]/(Fibonorial[2 n + 1] Fibonorial[n + 1]), {n, 1, 30}] (* The sequence itself *)
QPochhammer[-1/GoldenRatio^2] (* The Fibonacci factorial constant C in the asymptotic expansion *)

a(n) ~ 5*phi^(2*n^2 - 3*n - 2)/C where phi = (1+sqrt(5))/2, and C = (-1/phi^2; -1/phi^2)_inf is the Fibonacci factorial constant whose decimal expansion is given in A062073.

A276499 Decimal expansion of Fibonorial(1/2).

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Sep 05 2016

This constant can be thought of as A003266(1/2).

The fibonorial of (not necessarily an integer) x is defined as x!F = (phi^(x*(x+1)/2) / F(x+1)) * Product{n=1..inf} F(n+1)^(x+1)/(F(n)^x * F(x+n+1)), where F(x) = (phi^x - cos(Pi*x)/phi^x)/sqrt(5), where phi = (1+sqrt(5))/2 is the golden ratio. It satisfies the recurrence 0!_F = 1, x!_F = (x-1)!_F * F(x), and agrees with A003266(x) at integer points.

			0.98260982501326431122377480560574910946538...

Simon Plouffe, Fibonacci factorials.
Eric Weisstein's World of Mathematics, Fibonorial, Fibonacci Factorial Constant.

Cf. A003266, A062073.

RealDigits[N[GoldenRatio^(3/8) QPochhammer[-1/GoldenRatio^2]/5^(1/4), 100]][[1]]

Fibonorial(1/2) = phi^(3/8) * C / 5^(1/4), where C = A062073 is the Fibonacci factorial constant.

cp's OEIS Frontend

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

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A253267 Decimal expansion of a constant related to A152686.

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A062072 Continued fraction expansion of Fibonacci factorial constant.

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A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

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A191994 (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).

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PARI

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A382910 a(n) = A003266(n)^2.

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A094221 1/detM(n) where M(n) is the n X n matrix m(i,j)=F(i)/F(i+j-1) and F(i)=i-th Fibonacci number.

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A241990 Decimal expansion of 'delta', a constant arising in the asymptotics of the regularized product of the Fibonacci numbers.

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A271421 a(n) = fibonorial(3n)/(fibonorial(2n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).