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A144129 ChebyshevT(3, n). 15
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
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)