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

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

Offset: 0

Author

David Penney (david(AT)math.uga.edu)

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A015238, A032528, A153641.

Programs

Magma

[((2*n+3)*n^2 - (n mod 2))/4: n in [0..100]]; // G. C. Greubel, Nov 03 2024

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 *)

SageMath

[((2*n+3)*n^2 - (n%2))//4 for n in range(101)] # G. C. Greubel, Nov 03 2024

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

Extensions

More terms from Henry Bottomley, Nov 13 2000