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A011934 a(n) = |1^3 - 2^3 + 3^3 - 4^3 + ... + (-1)^(n+1)*n^3|. 13
0, 1, 7, 20, 44, 81, 135, 208, 304, 425, 575, 756, 972, 1225, 1519, 1856, 2240, 2673, 3159, 3700, 4300, 4961, 5687, 6480, 7344, 8281, 9295, 10388, 11564, 12825, 14175, 15616, 17152, 18785, 20519, 22356, 24300, 26353, 28519, 30800, 33200, 35721, 38367, 41140 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From the formula a(n) = n^3 - a(n-1) it follows that a(n-1) + a(n) = n^3. Thus the sum of two consecutive terms (call them the "former" and "latter" terms) is a cube of the index of the "latter" term. - Alexander R. Povolotsky, Jan 09 2008
The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus we get expression a(k) = |2^(-4)(P(3,1)-(-1)^k P(3,2k+1))|. - Peter Luschny, Jul 12 2009
a(n) is the number of (w,x,y} having all terms in {0,...,n} and w<floor((x+y)/2). Also, the number of (w,x,y} having all terms in {0,...,n} and w>=floor((x+y)/2). [Clark Kimberling, Jun 02 2012]
REFERENCES
Eldon Hansen's _A Table of Series and Products_ (Prentice-Hall, 1975) gives the sum in Formula 6.2.2 in terms of Euler polynomials.
LINKS
Kenneth B. Davenport, Problem 913, Pi Mu Epsilon Journal, Vol. 10, No. 6, Spring 1997, p. 492.
FORMULA
a(n) = |(1/8)*(-1 + (-1)^n - 6*(-1)^n*n^2 - 4*(-1)^n*n^3)|. - Henry Bottomley, Nov 13 2000
a(n) = n^3-a(n-1) = a(n-1)+A032528(n) = ceiling(A015238(n+1)/4) = ceiling((n+1)^2*(2n-1)/4). - Henry Bottomley, Nov 13 2000
G.f.: (x^3 + 4*x^2 + x)/(x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 1). - Alexander R. Povolotsky, Apr 26 2008
{-a(n)-a(n+1)+n^3+3*n^2+3*n+1, a(0)=0, a(1)=1, a(2)=7, a(3)=20}. - Robert Israel, May 14 2008
a(n) = Sum_{k=1..n} floor((2*n+1)*k/2). - Wesley Ivan Hurt, Apr 01 2017
MAPLE
a := n -> ((2*n+3)*n^2-(n mod 2))/4; # Peter Luschny, Jul 12 2009
MATHEMATICA
k=0; lst={k}; Do[k=n^3-k; AppendTo[lst, k], {n, 1, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(4 n^3 - 6 n^2 - (-1)^n + 1)/8, {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
Abs[Accumulate[Times@@@Partition[Riffle[Range[0, 50]^3, {1, -1}, {1, -1, 2}], 2]]] (* Harvey P. Dale, May 20 2019 *)
CROSSREFS
Sequence in context: A298488 A175428 A232599 * A159222 A100206 A298288
KEYWORD
nonn,easy
AUTHOR
David Penney (david(AT)math.uga.edu)
EXTENSIONS
More terms from Henry Bottomley, Nov 13 2000
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)