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A002593
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n^2*(2*n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.
(Formerly M5199 N2262)
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3
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0, 1, 28, 153, 496, 1225, 2556, 4753, 8128, 13041, 19900, 29161, 41328, 56953, 76636, 101025, 130816, 166753, 209628, 260281, 319600, 388521, 468028, 559153, 662976, 780625, 913276, 1062153, 1228528, 1413721, 1619100, 1846081
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The m-th term, for m = A065549(n), is perfect (A000396). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 04 2002
Partial sums of A016755. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 06 2004
Also, k-th triangular number, where k=2n^2 - 1=A056220(n), i.e. a(n)=A000217(A056220(n)). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 11 2004
Odd numbers and their squares both having the form 2x-+1, we may write (2r+1)^3=(2r+1)*(2s-1), where s=centered squares=(r+1)^2 + r^2. Since 2r+1=(r+1)^2 - r^2, it follows immediately from summing telescopingly over n-1, the product 2*{(r+1)^4 - r^4} - {(r+1)^2 - r^2}, that sum_{0, n-1} (2r+1)^3 = 2*n^4 - n^2 = n^2*(2n^2 - 1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 16 2004
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 169, #31.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. J. Zerger, Proof without words: The sum of consecutive odd cubes is a triangular number, Math. Mag., 68 (1995), 371.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
G. Xiao, Sigma Server, Operate on "(2*n-1)^3"
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FORMULA
| G.f.: (-x^4 - 23 x^3 - 23 x^2 - x)/(x - 1)^5 [From Harvey P. Dale, Mar 28 2011]
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MAPLE
| A002593:=-z*(z+1)*(z**2+22*z+1)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| s = 0; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 60, 2}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
CoefficientList[Series[(-x^4-23x^3-23x^2-x)/(x-1)^5, {x, 0, 80}], x] (* or *)
Table[n^2 (2n^2-1), {n, 0, 80}] (* From Harvey P. Dale, Mar 28 2011 *)
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PROG
| (MAGMA) [n^2*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Sep 07 2011
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CROSSREFS
| Cf. A002309.
Sequence in context: A069917 A028380 A188778 * A015881 A026910 A172220
Adjacent sequences: A002590 A002591 A002592 * A002594 A002595 A002596
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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