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
Index entries for linear recurrences with constant coefficients, signature (3, -2, -2, 3, -1).
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
KEYWORD
nonn,easy
AUTHOR
David Penney (david(AT)math.uga.edu)
EXTENSIONS
More terms from Henry Bottomley, Nov 13 2000
STATUS
approved