0,3

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

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.

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"

A002593:=-z*(z+1)*(z**2+22*z+1)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A002309.

Adjacent sequences: A002590 A002591 A002592 this_sequence A002594 A002595 A002596

Sequence in context: A042532 A069917 A028380 this_sequence A015881 A026910 A085377

nonn,nice,easy

njas