A016758 a(n) = (2*n+1)^6.

1, 729, 15625, 117649, 531441, 1771561, 4826809, 11390625, 24137569, 47045881, 85766121, 148035889, 244140625, 387420489, 594823321, 887503681, 1291467969, 1838265625, 2565726409, 3518743761, 4750104241, 6321363049, 8303765625, 10779215329, 13841287201, 17596287801

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A300709.

Programs

Magma

[(2*n+1)^6: n in [0..30]]; // Vincenzo Librandi, Sep 07 2011

Mathematica

(2*Range[0,20]+1)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1, 729,15625,117649,531441,1771561,4826809},20] (* Harvey P. Dale, Dec 26 2012 *)

PARI

vector(30, n, n--; (2*n+1)^6) \\ G. C. Greubel, Sep 15 2018

Formula

a(0)=1, a(1)=729, a(2)=15625, a(3)=117649, a(4)=531441, a(5)=1771561, a(6)=4826809, 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). - Harvey P. Dale, Dec 26 2012

G.f.: (1 +722*x +10543*x^2 +23548*x^3 +10543*x^4 +722*x^5 +x^6)/(1-x)^7 . - R. J. Mathar, Jul 07 2017

Sum_{n>=0} 1/a(n) = Pi^6/960 (A300709). - Amiram Eldar, Oct 10 2020

From Amiram Eldar, Jan 28 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2)*(cos(sqrt(3)*Pi/2) + cosh(Pi/2))/2.

Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(sqrt(3)*Pi/2)/24. (End)