|
%I #52 Sep 08 2022 08:45:28
%S 0,15,80,255,624,1295,2400,4095,6560,9999,14640,20735,28560,38415,
%T 50624,65535,83520,104975,130320,159999,194480,234255,279840,331775,
%U 390624,456975,531440,614655,707280,809999,923520,1048575,1185920,1336335
%N a(n) = n^4 - 1.
%C 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)).
%C A129292(n) = number of divisors of a(n) that are not greater than n. - _Reinhard Zumkeller_, Apr 09 2007
%H Vincenzo Librandi, <a href="/A123865/b123865.txt">Table of n, a(n) for n = 1..1000</a>
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: x^2*(15 + 5*x + 5*x^2 - x^3)/(1-x)^5. - _Colin Barker_, Jan 10 2012
%F -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
%F From _Vaclav Kotesovec_, Feb 14 2015: (Start)
%F Sum_{n>=2} 1/a(n) = 7/8 - Pi*coth(Pi)/4 = A256919.
%F Sum_{n>=2} (-1)^n / a(n) = 1/8 - Pi/(4*sinh(Pi)). (End)
%F a(n) = A005563(A005563(n)). - _Bruno Berselli_, May 28 2015
%F E.g.f.: 1 + (-1 + x + 7*x^2 + 6*x^3 + x^4)*exp(x). - _G. C. Greubel_, Aug 08 2019
%F Product_{n>=2} (1 + 1/a(n)) = 4*Pi*csch(Pi). - _Amiram Eldar_, Jan 20 2021
%p seq(n^4 -1, n=1..40); # _G. C. Greubel_, Aug 08 2019
%t Table[n^4-1,{n,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%o (Magma) [n^4 - 1: n in [1..40]]; // _Vincenzo Librandi_, May 01 2011
%o (PARI) vector(40, n, n^4 -1) \\ _G. C. Greubel_, Aug 08 2019
%o (Sage) [n^4 -1 for n in (1..40)] # _G. C. Greubel_, Aug 08 2019
%o (GAP) List([1..40], n-> n^4 -1); # _G. C. Greubel_, Aug 08 2019
%Y Cf. A000583, A005563, A024002, A123866, A123867, A123868, A219086.
%K nonn,easy
%O 1,2
%A _Reinhard Zumkeller_, Oct 16 2006
|
