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A144129 ChebyshevT(3, n). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

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 *)

PROG

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

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

(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

CROSSREFS

Sequence in context: A256645 A175549 A159541 * A026915 A136293 A065013

Adjacent sequences:  A144126 A144127 A144128 * A144130 A144131 A144132

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

STATUS

approved

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Last modified June 21 23:01 EDT 2018. Contains 305646 sequences. (Running on oeis4.)