A339605 -id:A339605 - cp's OEIS frontend

A058895 a(n) = n^4 - n.

0, 0, 14, 78, 252, 620, 1290, 2394, 4088, 6552, 9990, 14630, 20724, 28548, 38402, 50610, 65520, 83504, 104958, 130302, 159980, 194460, 234234, 279818, 331752, 390600, 456950, 531414, 614628, 707252, 809970, 923490, 1048544, 1185888, 1336302, 1500590, 1679580

Offset: 0

Henry Bottomley, Jan 08 2001

a(n) is the number of ways to assign 4 different students to n different dorm rooms, each of which can hold at most 3 students. In other words, a(n) is the number of functions f:[4]->[n] with the size of the pre-image set of each element of the codomain at most 3. - Dennis P. Walsh, Mar 21 2013

a(n) are the values of m that yield integer solutions to this family of equations: x = sqrt(m + sqrt(x)), which may also be viewed as an infinitely recursive radical. The real solutions for x at each m = a(n) is n^2, except at n = 1 (m = 0) where x = 0 or 1 is a solution. - Richard R. Forberg, Oct 15 2014

Harry J. Smith, Table of n, a(n) for n = 0..2000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A002061, A002378, A024001, A027444, A027482, A068601, A033562, A185065, A339605.

[n^4-n: n in [0..40]]; // Vincenzo Librandi, Feb 23 2012

seq(n*(n^3-1),n=0..25) ; # R. J. Mathar, Dec 10 2015

Table[n^4 - n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n) = { n^4-n } \\ Harry J. Smith, Jun 23 2009

(2*x^2*(7+4*x+x^2)/(1-x)^5).series(x, 37).coefficients(x, sparse=False) # Stefano Spezia, Jul 09 2021

a(n) = n*(n-1)*(n^2+n+1) = A000583(n) - n = A002061(n+1) * A002378(n-1) = (n-1) * A027444(n) = -n * A024001(n).
a(n) = 2*A027482(n). - Zerinvary Lajos, Jan 28 2008
a(n) = floor(n^7/(n^3+1)). - Gary Detlefs, Feb 11 2010
a(n)^3 = (a(n)/n)^4 + (a(n)/n)^3. - Vincenzo Librandi, Feb 23 2012
a(n)^3 + A068601(n)^3 + A033562(n)^3 = A185065(n)^3, for n > 0. - Vincenzo Librandi, Mar 13 2012
G.f.: 2*x^2*(7 + 4*x + x^2)/(1 - x)^5. - Colin Barker, Apr 23 2012
a(n) = 14*C(n,2) + 36*C(n,3) + 24*C(n,4). - Dennis P. Walsh, Mar 21 2013
Sum_{n>=2} (-1)^n/a(n) = (Pi/3)*sech(Pi*sqrt(3)/2) + 4*log(2)/3 - 1 = 0.06147271494... . - Amiram Eldar, Jul 04 2020
Sum_{n>=2} 1/a(n) = A339605. - R. J. Mathar, Jan 08 2021
E.g.f.: exp(x)*x^2*(7 + 6*x + x^2). - Stefano Spezia, Jul 09 2021
a(n) = 12*A000332(n+2) + 2*A000537(n-1). - Yasser Arath Chavez Reyes, Apr 05 2024

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

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

A339097 Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 24 2020

			0.0390670072379950810608...

Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).

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

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

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

Equals Sum_{k>=2} (k^3 -3*k^2 + k - 2)/(k^5 - k).
Equals 3/8 - 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 3/8 - Re(H(I))/2, where H is the harmonic number function.
Equals 1/4 - A338858.
Equals Sum_{k>=2} 1/(k*(k^4 - 1)). - Vaclav Kotesovec, Dec 24 2020

A339801 Decimal expansion of the real part of harmonic number H(1/2 + i*sqrt(3)/2), where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 17 2020

For imaginary part see A339802.

For real b, Im(Psi(1/2 + b*i)) = Pi*tanh(Pi*b)/2, but no such closed formula is known for the real part (see Wikipedia link). - Vaclav Kotesovec, Dec 19 2020

			0.862289106171836386535085450...

Wikipedia, Digamma function

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A339802.

RealDigits[N[Re[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Equals 1/2 + gamma + Re(Psi(1/2 + i*sqrt(3)/2)), where gamma is the Euler-Mascheroni constant (see A001620) and Psi is the digamma function.
Equals -1/2 + 3*A339604 + 3*A339606.
Equals Re((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

A339802 Decimal expansion of the imaginary part of harmonic number H(1/2 + i*sqrt(3)/2) where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 17 2020

For real part of H(1/2 + i*sqrt(3)/2) see A339801.

			0.691215820928755403365848...

Cf. A256919, A338815, A339135, A339529, A339530, A339605, A339605, A339606, A339801.

RealDigits[N[Im[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Equals (Pi/2)*tanh(Pi*sqrt(3)/2) - sqrt(3)/2.
Equals Im(Psi(3/2 + i*sqrt(3)/2)).
Equals -sqrt(3)/2 + Im(Psi(1/2 + i*sqrt(3)/2)).
Equals Im((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

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

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A058895 a(n) = n^4 - n.

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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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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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A339097 Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.

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A339801 Decimal expansion of the real part of harmonic number H(1/2 + i*sqrt(3)/2), where i=sqrt(-1).

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