A123865 a(n) = n^4 - 1.

0, 15, 80, 255, 624, 1295, 2400, 4095, 6560, 9999, 14640, 20735, 28560, 38415, 50624, 65535, 83520, 104975, 130320, 159999, 194480, 234255, 279840, 331775, 390624, 456975, 531440, 614655, 707280, 809999, 923520, 1048575, 1185920, 1336335

Offset: 1

Author

Reinhard Zumkeller, Oct 16 2006

Keywords

• nonn
• easy

Comments

a(n) mod 5 = 0 iff n mod 5 > 0: a(A008587(n)) = 4; a(A047201(n)) = 0; a(n) mod 5 = 4*(1-A079998(n)).

A129292(n) = number of divisors of a(n) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000583, A005563, A024002, A123866, A123867, A123868, A219086.

Programs

GAP

List([1..40], n-> n^4 -1); # G. C. Greubel, Aug 08 2019

Magma

[n^4 - 1: n in [1..40]]; // Vincenzo Librandi, May 01 2011

Maple

seq(n^4 -1, n=1..40); # G. C. Greubel, Aug 08 2019

Mathematica

Table[n^4-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

PARI

vector(40, n, n^4 -1) \\ G. C. Greubel, Aug 08 2019

Sage

[n^4 -1 for n in (1..40)] # G. C. Greubel, Aug 08 2019

Formula

G.f.: x^2*(15 + 5*x + 5*x^2 - x^3)/(1-x)^5. - Colin Barker, Jan 10 2012

-4*a(n+1) = -4*n*(n+2)*(n^2+2*n+2) = (n+n*i)*(n+2+n*i)*(n+(n+2)*i)*(n+2+(n+2)*i), where i is the imaginary unit. - Jon Perry, Feb 05 2014

From Vaclav Kotesovec, Feb 14 2015: (Start)

Sum_{n>=2} 1/a(n) = 7/8 - Pi*coth(Pi)/4 = A256919.

Sum_{n>=2} (-1)^n / a(n) = 1/8 - Pi/(4*sinh(Pi)). (End)

a(n) = A005563(A005563(n)). - Bruno Berselli, May 28 2015

E.g.f.: 1 + (-1 + x + 7*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Aug 08 2019

Product_{n>=2} (1 + 1/a(n)) = 4*Pi*csch(Pi). - Amiram Eldar, Jan 20 2021