A090986 -id:A090986 - cp's OEIS frontend

A136008 a(n) = n^6 - n^2.

0, 0, 60, 720, 4080, 15600, 46620, 117600, 262080, 531360, 999900, 1771440, 2985840, 4826640, 7529340, 11390400, 16776960, 24137280, 34011900, 47045520, 63999600, 85765680, 113379420, 148035360, 191102400, 244140000, 308915100

Offset: 0

Rolf Pleisch, Mar 16 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A001014, A013661, A072691, A090986.

[6*Binomial(n^2 +1, 3): n in [0..30]]; // G. C. Greubel, Feb 07 2022

f[n_]:=n^6-n^2; f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,60,720,4080,15600,46620},30] (* Harvey P. Dale, Dec 07 2024 *)

[n^2*(n^4-1) for n in range(0,31)] # Zerinvary Lajos, Jul 16 2008

G.f.: 60*x^2*(1 +5*x +5*x^2 +x^3)/(1-x)^7. - Alexander R. Povolotsky, Apr 01 2008
a(n) = A001014(n) - A000290(n). - Omar E. Pol, Dec 26 2008
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=2} 1/a(n) = 7/8 - Pi^2/6 + Pi*coth(Pi)/4.
Sum_{n>=2} (-1)^n/a(n) = -7/8 + Pi^2/12 + Pi*csch(Pi)/4 = -7/8 + A072691 + (1/4) * A090986. (End)
a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7). - Wesley Ivan Hurt, May 04 2021
From G. C. Greubel, Feb 07 2022: (Start)
a(n) = 6*binomial(n^2 + 1, 3).
E.g.f.: x^2*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x). (End)

Extended by Ray Chandler, Dec 13 2008

A144668 Decimal expansion of product_{n=2..infinity} (n^9-1)/(n^9+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.99599126589340465511919...

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

Cf. A090986.

p = 2/3*Product[Gamma[2 + (-1)^(k + k/9)], {k, {1, 2, 4, 5, 7, 8}}]/ Product[Gamma[2 - (-1)^(k + k/9)], {k, {1, 2, 4, 5, 7, 8}}]; RealDigits[Re[p], 10, 105][[1]] (* Jean-François Alcover, Feb 11 2013, updated Nov 18 2015 *)

More terms from Jean-François Alcover, Feb 11 2013

A144669 Decimal expansion of product_{n=2..infinity} (n^10-1)/(n^10+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.99801282617298278419003981450896856553...

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

Cf. A090986.

f[k_] := Sin[(-1)^(k/10)*Pi]; RealDigits[Pi/(5*Sinh[Pi])*Product[f[k], {k, {2, 4, 6, 8}}]/ Product[f[k], {k, {1, 3, 7, 9}}] // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

More terms from Jean-François Alcover, Feb 12 2013

A330864 Decimal expansion of sinh(Pi/2)/2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

This constant is transcendental.

			(1 + 1/2^2) * (1 - 1/3^2) * (1 + 1/4^2) * (1 - 1/5^2) * (1 + 1/6^2) * ... = (e^(Pi/2) - e^(-Pi/2))/4 = 1.15064945115364743673152001...

Index entries for transcendental numbers

Cf. A042972, A049006, A090986, A156648, A175615, A308715, A308716, A308718, A323983, A330865, A334401.

Mathematica
```
RealDigits[Sinh[Pi/2]/2, 10, 110] [[1]]
```

sinh(Pi/2)/2 \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=1} Pi^(2*k-1)/(4^k*(2*k-1)!).
Equals Product_{k>=2} (1 + (-1)^k/k^2).
Equals (i^(-i) - i^i)/4, where i is the imaginary unit.

A352527 Decimal expansion of Sum_(k>=1) (-1)^k * zeta(2k)/(2k) (negated).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 19 2022

			-0.65092319930185633888521683150394766...

Les-mathematiques.net, Somme arctan(1/k).
Cornel Ioan Vălean, Problem 11924, The American Mathematical Monthly, Vol. 123, No. 7 (2016), p. 722; An Integral with Fractional Part of Tangent, Solution to Problem 11924 by Edward and Roberta White, ibid., Vol. 125, No. 5 (2018), pp. 468-470.

Cf. A013661, A002117, A013662, A090986.

Maple
```
evalf(log(Pi/sinh(Pi)) / 2, 100);
```

RealDigits[Log[Pi/Sinh[Pi]]/2, 10, 100][[1]] (* Amiram Eldar, Mar 19 2022 *)

Equals log(Pi/sinh(Pi)) / 2.

Equals Integral_{x=0..Pi/2} ({tan(x)}/tan(x) - 1) dx, where {x} = x - floor(x) is the fractional part of x (Vălean, 2016). - Amiram Eldar, Feb 08 2024

A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 12 2023

For Re(Gamma(-A366545 + i*sqrt(1-A366545^2))) = -0.930840199... see A366545.

			0.9565130903466545656...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A364821, A365317, A365318, A366345.

xmin = Re[x /. FindRoot[1/(2 Sqrt[1 - x^2]) I (Gamma[1 + x - I Sqrt[1 - x^2]] PolyGamma[0, x - I Sqrt[1 - x^2]] - Gamma[1 + x + I Sqrt[1 - x^2]] PolyGamma[0,
         x + I Sqrt[1 - x^2]]), {x, -0.98}, WorkingPrecision -> 110]];
 RealDigits[xmin, 10, 106][[1]]

A367976 Decimal expansion of Sum_{k >= 0} (-1)^k/(1+k^2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 07 2023

			0.636014527491066581475118291836...

Cf. A113319.

1/4*(2-Pi*tanh(Pi/2)+Pi*coth(Pi/2)) ; evalf(%) ;

RealDigits[(1 + Pi*Csch[Pi])/2, 10, 120][[1]] (* Amiram Eldar, Dec 11 2023 *)

sumalt(k=0, (-1)^k/(1+k^2)) \\ Michel Marcus, Dec 07 2023

Equals (2-Pi*tanh(Pi/2)+Pi*coth(Pi/2))/4 = (1 - A228048 + Pi/2*A367961)/2.
From Amiram Eldar, Dec 11 2023: (Start)
Equals (1 + Pi/sinh(Pi))/2.
Equals Integral_{x>=0} (cos(x)/cosh(x))^2 dx. (End)
Equals (1+A090986)/2. - R. J. Mathar, Dec 13 2023

A345929 Decimal expansion of Product_{k>=1} (1 + 1/k + 1/k^2)^2/(1 + 2/k + 3/k^2).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 29 2021

			1.84893618285824448522492685552083905909259253189639...

Jonathan M. Borwein and Robert M. Corless, Emerging Tools for Experimental Mathematics, The American Mathematical Monthly, Vol. 106, No. 10 (1999), pp. 889-909.

Cf. A090986.

RealDigits[3*Sqrt[2]*Cosh[Pi*Sqrt[3]/2]^2/(Pi*Sinh[Pi*Sqrt[2]]), 10, 100][[1]]

Equals 3*sqrt(2)*cosh(Pi*sqrt(3)/2)^2/(Pi*sinh(Pi*sqrt(2))).

cp's OEIS Frontend

A136008 a(n) = n^6 - n^2.

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A144668 Decimal expansion of product_{n=2..infinity} (n^9-1)/(n^9+1).

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A144669 Decimal expansion of product_{n=2..infinity} (n^10-1)/(n^10+1).

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A330864 Decimal expansion of sinh(Pi/2)/2.

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A352527 Decimal expansion of Sum_(k>=1) (-1)^k * zeta(2k)/(2k) (negated).

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A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

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Mathematica

A367976 Decimal expansion of Sum_{k >= 0} (-1)^k/(1+k^2).

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PARI

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A345929 Decimal expansion of Product_{k>=1} (1 + 1/k + 1/k^2)^2/(1 + 2/k + 3/k^2).

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