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,2*k+1)). Thus we get expression a(k) = |2^(-4)*(P(3,1)-(-1)^k P(3,2*k+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*(2*n-1)/4). - Henry Bottomley, Nov 13 2000
G.f.: x*(1 + 4*x + x^2)/(1 - 3*x + 2*x^2 + 2*x^3 - 3*x^4 + x^5). - Alexander R. Povolotsky, Apr 26 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
Table[(4*n^3 -6*n^2 +1-(-1)^n)/8, {n, 0, 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 *)
PROG
(Magma) [((2*n+3)*n^2 - (n mod 2))/4: n in [0..100]]; // G. C. Greubel, Nov 03 2024
(SageMath) [((2*n+3)*n^2 - (n%2))//4 for n in range(101)] # G. C. Greubel, Nov 03 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Penney (david(AT)math.uga.edu)
EXTENSIONS
More terms from Henry Bottomley, Nov 13 2000
STATUS
approved