|
%I #53 Apr 09 2024 17:19:57
%S 0,1,26,99,244,485,846,1351,2024,2889,3970,5291,6876,8749,10934,13455,
%T 16336,19601,23274,27379,31940,36981,42526,48599,55224,62425,70226,
%U 78651,87724,97469,107910,119071,130976,143649,157114,171395,186516
%N a(n) = ChebyshevT(3, n).
%C The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=3, thus a(k) = |(P(3,0)-(-1)^k*P(3,2*k))/2|. - _Peter Luschny_, Jul 12 2009
%C Partial sums of A069190. - _J. M. Bergot_, Jul 13 2013
%H Vincenzo Librandi, <a href="/A144129/b144129.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 4*n^3 - 3*n. - _Klaus Brockhaus_, Jan 11 2009
%F G.f.: x*(1 + 22*x + x^2)/(1 - x)^4. - _Klaus Brockhaus_, Jan 11 2009
%F a(n) = cosh(3*arccosh(n)) = cos(3*arccos(n)). - _Artur Jasinski_, Feb 14 2010
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 30 2012
%F a(n) = 24*A000292(n-1) + n. - _Bruce J. Nicholson_, Jun 12 2020
%F From _Gerry Martens_, Apr 06 2024: (Start)
%F a(n) = Imaginary part of -(1/2)*(2*n*i-1)^3.
%F a(n) = -4*(1/4 + n^2)^(3/2)*sin(3*arctan(2*n)). (End)
%p a := n -> (4*n^2-3)*n; # _Peter Luschny_, Jul 12 2009
%t lst={};Do[AppendTo[lst,ChebyshevT[3,n]],{n,0,10^2}];lst
%t Round[Table[N[Cosh[3 ArcCosh[n]], 100], {n, 0, 20}]] (* _Artur Jasinski_, Feb 14 2010 *)
%t CoefficientList[Series[x*(1+22*x+x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 30 2012 *)
%t LinearRecurrence[{4,-6,4,-1},{0,1,26,99},40] (* _Harvey P. Dale_, Apr 02 2015 *)
%o (Magma) [ 4*n^3-3*n: n in [0..36] ]; // _Klaus Brockhaus_, Jan 11 2009
%o (PARI) a(n) = 4*n^3-3*n \\ _Charles R Greathouse IV_, Feb 08 2012
%o (Magma) I:=[0, 1, 26, 99]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 30 2012
%Y Cf. A000292, A069190, A153641.
%K nonn,easy
%O 0,3
%A _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008
|
