A024006 -id:A024006 - cp's OEIS frontend

A339530 Decimal expansion of Sum_{k>=1} (zeta(8*k)-1).

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

Offset: 0

Vaclav Kotesovec, Dec 08 2020, following a suggestion of Artur Jasinski

			0.00409269829928628730747620468964025986524982473540016981249105600555721...

Cf. A024006, A256919, A339529.

Join[{0, 0}, RealDigits[15/16 - Pi*Coth[Pi]/8 + Pi*(Sin[Sqrt[2]*Pi] + Sinh[Sqrt[2]*Pi]) / (4*Sqrt[2]*(Cos[Sqrt[2]*Pi] - Cosh[Sqrt[2]*Pi])), 10, 100][[1]]]

Equals Sum_{k>=2} 1/(k^8 - 1).
Equals 15/16 - Pi*coth(Pi)/8 + Pi * (sin(sqrt(2)*Pi) + sinh(sqrt(2)*Pi)) / (4*sqrt(2) * (cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))).
Equals (1/2)*Sum_{k>=2} 1/(k^4-1) - (1/2)*Sum_{k>=2} 1/(k^4+1) = (A256919-A256920)/2. - R. J. Mathar, Jan 22 2021

A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 09 2020

For additional comments and generalization see attached text file.

			0.221689395109267...

Artur Jasinski, Sums of zeta(p*n+q)-1
T. J. Stieltjes, Table des valeurs des sommes Sk, Acta Math., Volume 10 (1887), 299-302.

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339605, A339606.

RealDigits[Chop[N[Sum[Zeta[3 n] - 1, {n, 1, Infinity}], 105]]][[1]]

suminf(k=1, zeta(3*k)-1) \\ Michel Marcus, Dec 09 2020

Equals Sum_{k>=2} 1/(k^3-1).
Equals 1 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals 1 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
Equals 7/6 - Pi*tanh(Pi*sqrt(3)/2)/(2*sqrt(3)) - A339605/2.
Equals 4/3 - Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) + A339606.
Equals 1 - A339605 - A339606.

A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 09 2020

			0.6865033423386238859646...

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339604, A339605.

evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120);  # Alois P. Heinz, Dec 12 2020

RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]

suminf(k=0, zeta(3*k+2)-1) \\ Michel Marcus, Dec 09 2020

Equals Sum_{k>=1} 1/(k^3 + 1).
Equals -1/3 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is digamma function, gamma is Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals -1/3 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
Equals 1 - A339605 - A339604.
Equals 1/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(3*k)-1). - Amiram Eldar, Jan 07 2024

A059830 a(n) = n^6 + n^4 + n^2 + 1.

Original entry on oeis.org

1, 4, 85, 820, 4369, 16276, 47989, 120100, 266305, 538084, 1010101, 1786324, 3006865, 4855540, 7568149, 11441476, 16843009, 24221380, 34117525, 47176564, 64160401, 85961044, 113614645, 148316260, 191435329, 244531876, 309373429, 387952660, 482505745, 595531444

Offset: 0

N. J. A. Sloane, Feb 25 2001

Harry J. Smith, 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. A059839.

Cf. A002522, A002523, A024006, A067998.

Table[(n^2+1)*(n^4+1),{n,0,40}] (* Alexander Adamchuk, Apr 13 2006 *)

a(n) = { my(f=n^2); (f + 1)*(f^2 + 1) } \\ Harry J. Smith, Jun 29 2009

a(n) = (n^2+1)*(n^4+1) = A002522(n)*A002523(n) = A002522(n)*A002522(n^2). a(n) = (n^8-1)/(n^2-1) = -A024006(n)/A067998(n+1), n>1. - Alexander Adamchuk, Apr 13 2006

G.f.: -(4*x^6+57*x^5+309*x^4+274*x^3+78*x^2-3*x+1)/(x-1)^7. - Colin Barker, Nov 05 2012

A024007 a(n) = 1 - n^9.

Original entry on oeis.org

1, 0, -511, -19682, -262143, -1953124, -10077695, -40353606, -134217727, -387420488, -999999999, -2357947690, -5159780351, -10604499372, -20661046783, -38443359374, -68719476735, -118587876496, -198359290367

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..388

Cf. A024004, A024006.

[1-n^9: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1 - n^9, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

A258837 a(n) = 1 - n^2.

Original entry on oeis.org

1, 0, -3, -8, -15, -24, -35, -48, -63, -80, -99, -120, -143, -168, -195, -224, -255, -288, -323, -360, -399, -440, -483, -528, -575, -624, -675, -728, -783, -840, -899, -960, -1023, -1088, -1155, -1224, -1295, -1368, -1443, -1520, -1599, -1680, -1763, -1848

Offset: 0

Vincenzo Librandi, Jun 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067998, A131386, A007531.

Sequences of the type 1-n^k: A024000 (k=1), this sequence (k=2), A024001 (k=3), A024002 (k=4), A024003 (k=5), A024004 (k=6), A024005 (k=7), A024006 (k=8), A024007 (k=9), A024008 (k=10), A024009 (k=11), A024010 (k=12).

Magma
```
[1-n^2: n in [0..50]];
```

I:=[1,0,-3]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Table[1 - n^2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 0, -3}, 50]

my(x='x+O('x^50)); Vec((1-3*x)/(1-x)^3) \\ G. C. Greubel, May 11 2017

G.f.: (1-3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = -A067998(n+1). - Joerg Arndt, Jun 13 2015
a(n) = (-1)^n*A131386(n+1). - Bruno Berselli, Jun 15 2015
E.g.f.: (1 - x - x^2)*exp(x). - G. C. Greubel, May 11 2017
Sum_{n>=2} 1/a(n) = -3/4. - Amiram Eldar, Feb 17 2023

A338858 Decimal expansion of Sum_{k>=0} (zeta(4*k+3)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 24 2020

For additional comments and generalization see A339604.

			0.2109329927620049189391952864...

Cf. A256919 (4*k), A339097 (4*k+1), A339083 (4k+2).

Cf. A024006, A338815, A339135, A339529, A339530, A339604, A339605, A339606.

RealDigits[N[Re[Sum[Zeta[4 n + 3] - 1, {n, 0, Infinity}]], 105]][[1]]

suminf(k=0, zeta(4*k+3)-1) \\ Michel Marcus, Dec 24 2020

Equals Sum_{k>=2} k/(k^4-1).
Equals -1/8 + gamma/2 + Re(Psi(i))/2, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals -1/8 + Re(H(I))/2, where H is the harmonic number function.

A339083 Decimal expansion of Sum_{k>=0} (zeta(4*k+2)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 24 2020

Sum_{k>=1} zeta(4*k)-1 see A256919.
Sum_{k>=1} zeta(4*k+1)-1 see A339097.
Sum_{k>=0} zeta(4*k+3)-1 see A338858.
For additional comments and generalization see A339604.

			0.663337023734290587067025397375...

Cf. A024006, A256919, A338815, A338858, A339097, A339135, A339529, A339530, A339604, A339605, A339605.

RealDigits[N[Sum[Zeta[4 n + 2] - 1, {n, 0, Infinity}], 105]][[1]]

suminf(k=0, zeta(4*k+2)-1) \\ Michel Marcus, Dec 24 2020

Equals Sum_{k>=2} k^2/(k^4-1).

Equals -1/8 + Pi*coth(Pi)/4 = -1/8 + A338815 = 3/4 - A256919.

a(104) corrected and more terms from Georg Fischer, Jun 06 2024

A258809 a(n) = n^8 - 1.

Original entry on oeis.org

0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, 99999999, 214358880, 429981695, 815730720, 1475789055, 2562890624, 4294967295, 6975757440, 11019960575, 16983563040, 25599999999, 37822859360, 54875873535, 78310985280, 110075314175

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A024006, A060890.

Cf. similar sequences listed in A258807.

Magma
```
[n^8-1: n in [1..40]];
```

Table[n^8 - 1, {n, 33}] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720}, 40]

G.f.: x^2*(225 + 4535*x + 14595*x^2 + 18069*x^3 + 569*x^4 + 3999*x^5 - 2511*x^6 + 1079*x^7 - 270*x^8 + 30*x^9) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
a(n) = (n - 1)*(n + 1)*(n^2 + 1)*(n^4 + 1) = -A024006(n). [Bruno Berselli, Jun 12 2015]

A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(3)/2)/2.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 11 2022

Imaginary part of psi(-1/2 + i*sqrt(3)/2) where psi is the digamma function.

			2.423266628497632696893294495...

Cf. A024006, A100554, A113319, A256919, A259171, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A347292.

RealDigits[N[Sqrt[3]/2 + 1/2 Pi Tanh[Sqrt[3] Pi/2], 105]][[1]]

imag(psi(-1/2+I*sqrt(3)/2)) \\ Michel Marcus, Feb 11 2022

Equals sqrt(3)*(1 - gamma/3 - Re(psi(-1/2 + i*sqrt(3)/2))/3 + A339606).

Last two digits corrected by Georg Fischer, May 15 2024

cp's OEIS Frontend

A339530 Decimal expansion of Sum_{k>=1} (zeta(8*k)-1).

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A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).

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A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

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A059830 a(n) = n^6 + n^4 + n^2 + 1.

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A024007 a(n) = 1 - n^9.

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Magma

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A258837 a(n) = 1 - n^2.

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Magma

Magma

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A338858 Decimal expansion of Sum_{k>=0} (zeta(4*k+3)-1).

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A339083 Decimal expansion of Sum_{k>=0} (zeta(4*k+2)-1).

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A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(3)/2)/2.