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Search: id:A002593
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| A002593 |
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n^2*(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.
(Formerly M5199 N2262)
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+0
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; listen)
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OFFSET
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0,3
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COMMENT
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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
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
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.
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
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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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MAPLE
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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
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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]
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CROSSREFS
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Cf. A002309.
Sequence in context: A042532 A069917 A028380 this_sequence A015881 A026910 A085377
Adjacent sequences: A002590 A002591 A002592 this_sequence A002594 A002595 A002596
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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