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A011934 a(n) = |1^3 - 2^3 + 3^3 - 4^3 + ... + (-1)^(n+1)*n^3|. 12
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

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Kenneth B. Davenport, Problem 913, Pi Mu Epsilon Journal, Vol. 10, No. 6, Spring 1997, p. 492.

Skidmore College Problem Group, Solution to Problem #913 from the Pi Mu Epsilon Journal

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 *)

CROSSREFS

Sequence in context: A298488 A175428 A232599 * A159222 A100206 A298288

Adjacent sequences:  A011931 A011932 A011933 * A011935 A011936 A011937

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 June 21 23:05 EDT 2018. Contains 305646 sequences. (Running on oeis4.)