A226735 -id:A226735 - cp's OEIS frontend

A016779 a(n) = (3*n + 1)^3.

1, 64, 343, 1000, 2197, 4096, 6859, 10648, 15625, 21952, 29791, 39304, 50653, 64000, 79507, 97336, 117649, 140608, 166375, 195112, 226981, 262144, 300763, 343000, 389017, 438976, 493039, 551368, 614125, 681472, 753571, 830584, 912673, 1000000, 1092727, 1191016

Offset: 0

N. J. A. Sloane

The inverse binomial transform is 1, 63, 216, 162, 0, 0, 0 (0 continued). R. J. Mathar, May 07 2008

Perfect cubes with digital root 1 in base 10. Proof: perfect cubes are one of (3*s)^3, (3*s+1)^3 or (3*s+2)^3. Digital roots of (3*s)^3 are 0, digital roots of (3*s+1)^3 are 1, and digital roots of (3*s+2)^3 are 8, using trinomial expansion and the multiplicative property of digits roots. - R. J. Mathar, Jul 31 2010

			a(2) = (3*2+1)^3 = 343.
a(6) = (3*6+1)^3 = 6859.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3.
Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016791, A054966, A226735.

[(3*n+1)^3: n in [0..30]]; // Vincenzo Librandi, May 09 2011

Table[(3n+1)^3,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,64,343,1000},40] (* Harvey P. Dale, Oct 31 2016 *)

a(n)=(3*n+1)^3 \\ Charles R Greathouse IV, Jan 02 2012

Sum_{n>=0} 1/a(n) = 2*Pi^3 / (81*sqrt(3)) + 13*zeta(3)/27.
O.g.f.: (1 + 60*x + 93*x^2 + 8*x^3)/(1 - x)^4. - R. J. Mathar, May 07 2008
E.g.f.: (1 + 63*x + 108*x^2 + 27*x^3)*exp(x). - Ilya Gutkovskiy, Jun 16 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
Sum_{n>=1} (-1)^n/a(n) = A226735. - R. J. Mathar, Feb 07 2024

A262178 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+1)^2.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Sep 14 2015

Also, decimal expansion of Sum_{h>=0} Sum_{j=0..h} (-1)^j*binomial(h, j)/(4*(1 + h)*(1 + 6*j)*(2 + 3*j)).

			1 - 1/16 + 1/49 - 1/100 + 1/169 - 1/256 + 1/361 - 1/484 + ...
0.9515177134164150418664828314727415315447285082326970513300324315296113...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A006752.
Cf. A113476: Sum_{k>=0} (-1)^k/(3*k+1).
Cf. A226735: Sum_{k>=0} (-1)^k/(3*k+1)^3.

RealDigits[(Zeta[2, 1/6] - Zeta[2, 2/3])/36, 10, 100][[1]]

sumalt(k=0, (-1)^k/(3*k+1)^2) \\ Michel Marcus, Sep 14 2015

zetahurwitz(2,1/6)/36 - zetahurwitz(2,2/3)/36 \\ Charles R Greathouse IV, Jan 31 2018

Equals (zeta(2, 1/6) - zeta(2, 2/3))/36, where zeta(s,a) is the Hurwitz zeta function.

A370000 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+2)^3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 07 2024

			1/8 - 1/125 + 1/512 - 1/1331 + ... = 0.118438778425057529625616861943025438732887982...

R. W. Gosper, Acceleration of Series, Artificial Intelligence (1974) # 304.

Cf. A226735, A016791.

5*Pi^3/2/3^(9/2)-13*Zeta(3)/36 ; evalf(%) ;

RealDigits[5*Pi^3/(2*3^(9/2)) - 13*Zeta[3]/36, 10, 120][[1]] (* Amiram Eldar, Feb 09 2024 *)

sumalt(k=0, (-1)^k/(3*k+2)^3) \\ Michel Marcus, Feb 07 2024

Equals Sum_{n>=0} (-1)^n/A016791(n).

Equals A226735 - 13*zeta(3)/18 = 5*Pi^3/(2*3^(9/2)) - 13*zeta(3)/36.

cp's OEIS Frontend

A016779 a(n) = (3*n + 1)^3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A262178 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+1)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A370000 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+2)^3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula