Search: id:A002593
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%I A002593 M5199 N2262
%S A002593 0,1,28,153,496,1225,2556,4753,8128,13041,19900,29161,41328,56953,
%T A002593 76636,101025,130816,166753,209628,260281,319600,388521,468028,
%U A002593 559153,662976,780625,913276,1062153,1228528,1413721,1619100,1846081
%N A002593 n^2*(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.
%C A002593 The m-th term, for m = A065549(n), is perfect (A000396). - Lekraj Beedassy 
               (blekraj(AT)yahoo.com), Jun 04 2002
%C A002593 Partial sums of A016755. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 
               06 2004
%C A002593 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
%C A002593 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
%D A002593 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002593 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002593 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.
%D A002593 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 169, #31.
%D A002593 F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., 
               Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
%D A002593 L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
%D A002593 M. J. Zerger, Proof without words: The sum of consecutive odd cubes is 
               a triangular number, Math. Mag., 68 (1995), 371.
%H A002593 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002593 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002593 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">
               Operate on "(2*n-1)^3"</a>
%p A002593 A002593:=-z*(z+1)*(z**2+22*z+1)/(z-1)**5; [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%t A002593 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]
%Y A002593 Cf. A002309.
%Y A002593 Sequence in context: A042532 A069917 A028380 this_sequence A015881 A026910 
               A085377
%Y A002593 Adjacent sequences: A002590 A002591 A002592 this_sequence A002594 A002595 
               A002596
%K A002593 nonn,nice,easy
%O A002593 0,3
%A A002593 N. J. A. Sloane (njas(AT)research.att.com).

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