A016755 Odd cubes: a(n) = (2*n + 1)^3.

1, 27, 125, 343, 729, 1331, 2197, 3375, 4913, 6859, 9261, 12167, 15625, 19683, 24389, 29791, 35937, 42875, 50653, 59319, 68921, 79507, 91125, 103823, 117649, 132651, 148877, 166375, 185193, 205379, 226981, 250047, 274625, 300763, 328509, 357911, 389017, 421875

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Partial sums of A010014. - Jani Melik, May 20 2013

Terms end in the repeating sequence 1, 7, 5, 3, 9, ... - Melvin Peralta, Jul 08 2015

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Marc Chamberland and Armin Straub, On gamma quotients and infinite products, Advances in Applied Mathematics, Vol. 51, No. 5 (2013), pp. 546-562.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578, A005408, A010014, A016743, A153071, A154537.

Cf. A000330, A005917, A069074.

Programs

Magma

[(2*n+1)^3: n in [0..50]]; // Vincenzo Librandi, Sep 05 2011

Mathematica

Range[1,101,2]^3 (* Harvey P. Dale, Nov 18 2013 *)

PARI

a(n)=(2*n+1)^3 \\ Charles R Greathouse IV, Jan 02 2012

Python

def a(n): return (2*n+1)**3
print([a(n) for n in range(38)]) # Michael S. Branicky, Jan 27 2021

Formula

Sum_{n >= 0} 1/a(n) = 7 * zeta(3) / 8.

G.f.: (1+23*x+23*x^2+x^3)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 02 2012

a(n) = A000578(A005408(n)). - Michel Marcus, Jul 09 2015

E.g.f.: exp(x)*(1 + 26*x + 36*x^2 + 8*x^3). See A154537, row n=3. - Wolfdieter Lang, Mar 12 2017

From Bruce J. Nicholson, Dec 08 2019: (Start)

a(n) = 24 * A000330(n) + A005408(n).

a(n) = 2 * A005917(n+1) - A005408(n). (End)

Sum_{n>=0} (-1)^n/a(n) = Pi^3/32 (A153071). - Amiram Eldar, Oct 10 2020

Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/12)*(1 + sqrt(2)*cosh(sqrt(3)*Pi/4)) (Chamberland and Straub, 2013). - Amiram Eldar, Jan 26 2024