A144129 a(n) = ChebyshevT(3, n).

0, 1, 26, 99, 244, 485, 846, 1351, 2024, 2889, 3970, 5291, 6876, 8749, 10934, 13455, 16336, 19601, 23274, 27379, 31940, 36981, 42526, 48599, 55224, 62425, 70226, 78651, 87724, 97469, 107910, 119071, 130976, 143649, 157114, 171395, 186516

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Keywords

• nonn
• easy

Comments

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

Partial sums of A069190. - J. M. Bergot, Jul 13 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000292, A069190, A153641.

Programs

Magma

[ 4*n^3-3*n: n in [0..36] ]; // Klaus Brockhaus, Jan 11 2009

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

Maple

a := n -> (4*n^2-3)*n; # Peter Luschny, Jul 12 2009

Mathematica

lst={};Do[AppendTo[lst,ChebyshevT[3,n]],{n,0,10^2}];lst
Round[Table[N[Cosh[3 ArcCosh[n]], 100], {n, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)
CoefficientList[Series[x*(1+22*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,1,26,99},40] (* Harvey P. Dale, Apr 02 2015 *)

PARI

a(n) = 4*n^3-3*n \\ Charles R Greathouse IV, Feb 08 2012

Formula

a(n) = 4*n^3 - 3*n. - Klaus Brockhaus, Jan 11 2009

G.f.: x*(1 + 22*x + x^2)/(1 - x)^4. - Klaus Brockhaus, Jan 11 2009

a(n) = cosh(3*arccosh(n)) = cos(3*arccos(n)). - Artur Jasinski, Feb 14 2010

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012

a(n) = 24*A000292(n-1) + n. - Bruce J. Nicholson, Jun 12 2020

From Gerry Martens, Apr 06 2024: (Start)

a(n) = Imaginary part of -(1/2)*(2*n*i-1)^3.

a(n) = -4*(1/4 + n^2)^(3/2)*sin(3*arctan(2*n)). (End)