A090986 -id:A090986 - cp's OEIS frontend

A218131 Number of length 8 primitive (=aperiodic or period 8) n-ary words.

0, 0, 240, 6480, 65280, 390000, 1678320, 5762400, 16773120, 43040160, 99990000, 214344240, 429960960, 815702160, 1475750640, 2562840000, 4294901760, 6975673920, 11019855600, 16983432720, 25599840000, 37822664880, 54875639280, 78310705440, 110074982400

Offset: 0

Alois P. Heinz, Oct 21 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=8 of A143324.

Cf. A090986, A013662, A267315.

a:= n-> (n^4-1)*n^4:
seq(a(n), n=0..30);

Table[n^8 - n^4, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 30 2017 *)

G.f.: -240*x^2*(x+1)*(x^4+17*x^3+48*x^2+17*x+1)/(x-1)^9.
a(n) = n^8-n^4.
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=2} 1/a(n) = 15/8 - Pi^4/90 - Pi*coth(Pi)/4.
Sum_{n>=2} (-1)^n/a(n) = -7/8 + 7*Pi^4/720 - Pi*csch(Pi)/4 = -7/8 + A267315 - (1/4) * A090986. (End)

A364356 Decimal expansion of negative value of function Gamma(-A364355 + i*sqrt(1-A364355^2)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 08 2023

Only for x = A364355 = 0.54197987169489060244332278779... the Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1 (for one case is an imaginary number see A364821 and for other values x is a complex number).

			Gamma(-A364355 + i*sqrt(1-A364355^2)) = -0.6749332470449905963531...

Cf. A090986, A212877, A212878, A212880, A212879, A364355.

xmin=x /. FindRoot[Im[Gamma[-x + I Sqrt[1 - x^2]]], {x, 0.5}, WorkingPrecision -> 106];RealDigits[Re[-Gamma[-x + I Sqrt[1 - x^2]]/. x->xmin]][[1]]

A364821 Decimal expansion of the unique value of x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 07 2023

Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919... see A366345.

Also decimal expansion of the unique value of x in the range -1 < x < 1 for which the function Im(Gamma(x + i*sqrt(1-x^2)))/abs(Gamma(x + i*sqrt(1-x^2))) is minimized.

			0.149965974606491...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A366345.

RealDigits[
  x /. FindRoot[Re[Gamma[x + I Sqrt[1 - x^2]]], {x, 0.15},
    WorkingPrecision -> 106]][[1]]

Last digit corrected by Vaclav Kotesovec, Sep 02 2025

A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.8480540493529003921296501834...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
László Tóth, Transcendental Infinite Products Associated with the ±1 Thue-Morse Sequence, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.2.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A090986, A255434.

Digits := 120 :
m := 1:
for r from 2 to 10 do
omega := cos(Pi/r)+I*sin(Pi/r) :
x := (-1)^(m+1)*2*m*m!/r*mul( GAMMA(-m*omega^j)^(-(-1)^j),j=1..2*r-1) ;
x := Re(evalf(x)) ;
print(r,x) ;
od:

RealDigits[ -1/2*Pi*Csc[(-1)^(1/4)*Pi]*Csc[(-1)^(3/4)*Pi]*Sinh[Pi] // Re, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)
RealDigits[Re[N[Product[(n^4 - 1)/(n^4 + 1), {n, 2, Infinity}], 110]]][[1]] (* Bruno Berselli, Apr 02 2013 *)

Pi*sinh(Pi)/(cosh(Pi*sqrt(2))-cos(Pi*sqrt(2))) \\ Michel Marcus, Sep 07 2020

Equals Pi*sinh(Pi) / (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)). - Vaclav Kotesovec, Dec 08 2015

More terms from Jean-François Alcover, Feb 11 2013

A144664 Decimal expansion of Product_{n>=2} (n^5-1)/(n^5+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.92878693579955245375146991...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986, A255435.

p = 2*Gamma[2-(-1)^(1/5)] * Gamma[2+(-1)^(2/5)] * Gamma[2-(-1)^(3/5)] * Gamma[2+(-1)^(4/5)] / (Gamma[2+(-1)^(1/5)] * Gamma[2-(-1)^(2/5)] * Gamma[2+(-1)^(3/5)] * Gamma[2-(-1)^(4/5)]); RealDigits[Re[p], 10, 110][[1]] (* Jean-François Alcover, Feb 11 2013, updated Nov 18 2015 *)
RealDigits[Re[N[Product[(n^5 - 1)/(n^5 + 1), {n, 2, Infinity}], 110]]][[1]] (* Bruno Berselli, Apr 02 2013 *)

More terms from Jean-François Alcover, Feb 11 2013

A308718 Decimal expansion of Pi*sech(Pi/2)/2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jun 19 2019

			0.6260201656260738115441714982163444307583577855917...

Cf. A000290, A001844, A090986, A308714, A308717.

RealDigits[Pi Sech[Pi/2]/2, 10, 120][[1]]

Pi*(1/cosh(Pi/2))/2 \\ Michel Marcus, Jun 20 2019

Equals Product_{k>=1} 1/(1 - (-1)^k/k^2).
Equals Product_{k>=1} (1 - 1/(2*k*(k + 1) + 1)).
Equals Product_{k>=1} (1 - 1/A001844(k)).
Equals Integral_{x=0..oo} cos(x)/cosh(x) dx. - Amiram Eldar, Aug 10 2020

A144665 Decimal expansion of product_{n=2..infinity} (n^6-1)/(n^6+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.96590608576215921215...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986.

RealDigits[ -((Pi*(1 + Cosh[Sqrt[3]*Pi])*Csch[Pi])/(3*(Cos[Sqrt[3]*Pi] - Cosh[Pi]))), 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Pi*(cosh(sqrt(3)*Pi)+1)/sinh(Pi)/(cosh(Pi)-cos(sqrt(3)*Pi))/3 \\ - M. F. Hasler, Feb 23 2013

c=(Pi/3)*(cosh(sqrt(3)*Pi)+1)/(sinh(Pi)*(cosh(Pi)-cos(sqrt(3)*Pi))).

A144666 Decimal expansion of product_{n=2..infinity} (n^7-1)/(n^7+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.9834397805869282351144875535540...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986.

p = 2*Product[Gamma[2+(-1)^(k+k/7)], {k, 6}]/Product[Gamma[2-(-1)^(k+k/7)], {k, 6}]; RealDigits[Re[p], 10, 105][[1]] (* Jean-François Alcover, Feb 11 2013, updated Nov 18 2015 *)
RealDigits[Re[N[Product[(n^7 - 1)/(n^7 + 1), {n, 2, Infinity}], 110]]][[1]] (* Bruno Berselli, Apr 02 2013 *)

More terms from Jean-François Alcover, Feb 11 2013

A144667 Decimal expansion of product_{n=2..infinity} (n^8-1)/(n^8+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.9918784076582948169664...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986.

RealDigits[-1/4*Pi*Csc[(-1)^(1/8)*Pi]*Csc[(-1)^(3/8)*Pi]*Csc[(-1)^(5/8)*Pi] * Csc[(-1)^(7/8)*Pi]*Sin[(-1)^(1/4)*Pi]*Sin[(-1)^(3/4)*Pi]*Sinh[Pi] // Re, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Equals 2*A175619/A334411. - Vaclav Kotesovec, Apr 27 2020

More terms from Jean-François Alcover, Feb 11 2013

A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 07 2023

x = A364821 = 0.149965974606491089853... is the only real number x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

			Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919046...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A364821.

cp's OEIS Frontend

A218131 Number of length 8 primitive (=aperiodic or period 8) n-ary words.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A364356 Decimal expansion of negative value of function Gamma(-A364355 + i*sqrt(1-A364355^2)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364821 Decimal expansion of the unique value of x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A144664 Decimal expansion of Product_{n>=2} (n^5-1)/(n^5+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A308718 Decimal expansion of Pi*sech(Pi/2)/2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A144665 Decimal expansion of product_{n=2..infinity} (n^6-1)/(n^6+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A144666 Decimal expansion of product_{n=2..infinity} (n^7-1)/(n^7+1).

Views

Author

Keywords

Examples

Links

A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.