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Search: id:A000540
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| A000540 |
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Sum of 6th powers: 1^6 + 2^6 + ... + n^6.
(Formerly M5335 N2322)
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+0
12
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| 0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, 1978405, 3749966, 6735950, 11562759, 19092295, 30482920, 47260136, 71397705, 105409929, 152455810, 216455810, 302221931, 415601835, 563637724, 754740700, 998881325, 1307797101, 1695217590
(list; graph; listen)
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OFFSET
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0,3
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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.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
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FORMULA
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a(n) = n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42.
a(n) = Sqrt[Sum[Sum[(i*j)^6, {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
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MAPLE
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a:=n->sum (j^6, j=0..n): seq(a(n), n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007
A000540:=(z+1)*(z**4+56*z**3+246*z**2+56*z+1)/(z-1)**8; [Conjectured by S. Plouffe in his 1992 dissertation.]
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^6 od: seq(a[n], n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2008
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MATHEMATICA
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lst={}; s=0; Do[s=s+n^6; AppendTo[lst, s], {n, 10^2}]; lst..or..Table[Sum[k^6, {k, 1, n}], {n, 0, 100}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 14 2008]
s = 0; lst = {s}; Do[s += n^6; AppendTo[lst, s], {n, 1, 30, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
Accumulate[Range[0, 30]^6] [From Harvey P. Dale (hpd1(AT)nyu.edu), Jul 30 2009]
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PROGRAM
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(Other) sage: [bernoulli_polynomial(n, 7)/7 for n in xrange(1, 29)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
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CROSSREFS
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Cf. A101093.
Row 6 of array A103438.
Sequence in context: A013954 A116277 A008516 this_sequence A023875 A027463 A144500
Adjacent sequences: A000537 A000538 A000539 this_sequence A000541 A000542 A000543
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Significantly simpler and faster Mathematica program provided. [From Harvey P. Dale (hpd1(AT)nyu.edu), Jul 30 2009]
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