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A011934
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|1^3 - 2^3 + 3^3 - 4^3 + ... + (-1)^(n+1)*n^3|.
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8
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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; internal format)
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OFFSET
| 0,3
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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 (pevnev(AT)juno.com), Jan 09 2008
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)
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
a(k) = |2^(-4)(P(3,1)-(-1)^k P(3,2k+1))|. (End)
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REFERENCES
| Problem 913 of the Spring 1997 issue of the Pi Mu Epsilon Journal.
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.
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LINKS
| Skidmore College Problem Group, Solution to Problem #913 from the Pi Mu Epsilon Journal
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FORMULA
| (1/8)[ -1 + (-1)^n - 6*(-1)^n*n^2 - 4*(-1)^n*n^3 ]. - Henry Bottomley (se16(AT)btinternet.com), 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 (se16(AT)btinternet.com), 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 (pevnev(AT)juno.com), 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
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MAPLE
| a := n -> ((2*n+3)*n^2-(n mod 2))/4; [From Peter Luschny (peter(AT)luschny.de), Jul 12 2009]
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MATHEMATICA
| k=0; lst={k}; Do[k=n^3-k; AppendTo[lst, k], {n, 1, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
Table[(4 n^3 - 6 n^2 - (-1)^n + 1)/8, {n, 1, 100}] (* From Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
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CROSSREFS
| Sequence in context: A162024 A143058 A175428 * A159222 A100206 A007044
Adjacent sequences: A011931 A011932 A011933 * A011935 A011936 A011937
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KEYWORD
| nonn,easy
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AUTHOR
| David Penney (david(AT)math.uga.edu)
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EXTENSIONS
| More terms from Henry Bottomley (se16(AT)btinternet.com), Nov 13 2000
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