%I A000578 M4499 N1905
%S A000578 0,1,8,27,64,125,216,343,512,729,1000,1331,1728,2197,2744,3375,4096,4913,
%T A000578 5832,6859,8000,9261,10648,12167,13824,15625,17576,19683,21952,24389,
%U A000578 27000,29791,32768,35937,39304,42875,46656,50653,54872,59319,64000
%N A000578 The cubes: a(n) = n^3.
%C A000578 a(n) = sum of the next n odd numbers; i.e. group the odd numbers so that
the n-th group contains n elements like this (1), (3,5),(7,9,11),
(13,15,17,19),(21,23,25,27,29,),... then each group sum = n^3 = a(n).
Also the median of each group = n^2 = mean. As the sum of first n
odd numbers is n^2 this gives another proof of the fact that the
n-th partial sum = {n(n+1)/2}^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Sep 14 2002
%C A000578 Total number of triangles resulting from criss-crossing cevians within
a triangle so that two of its sides are each n-partitioned. - Lekraj
Beedassy (blekraj(AT)yahoo.com), Jun 02 2004
%C A000578 n^3 is the sum of the first n centered hexagonal numbers (A003215). -
Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 29 2004
%C A000578 Also structured triakis tetrahedral numbers (vertex structure 7) (Cf.
A100175 = alternate vertex); structured tetragonal prism numbers
(vertex structure 7) (Cf. A100177 = structured prisms); structured
hexagonal diamond numbers (vertex structure 7) (Cf. A100178 = alternate
vertex; A000447 = structured diamonds); and structured trigonal anti-diamond
numbers (vertex structure 7) (Cf. A100188 = structured anti-diamonds).
Cf. A100145 for more on structured polyhedral numbers . - James A.
Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A000578 Schlaefli symbol for this polyhedron: {4,3}
%C A000578 Least multiple of n such that every partial sum is a square. - Amarnath
Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
%C A000578 Draw a regular hexagon. Construct points on each side of the hexagon
such that these points divide each side into equally-sized segments
(i.e. a midpoint on each side or two points on each side placed to
divide each side into three equally-sized segments or so on), do
the same construction for every side of the hexagon so that each
side is equally divided in the same way. Connect all such points
to each other with lines that are parallel to at least one side of
the polygon. The result is a triangular tiling of the hexagon and
the creation of a number of smaller regular hexagons. The equation
gives the total number of regular hexagons found where n=the number
of points drawn+1. For example, if 1 point is drawn on each side
then n=1+1=2 and a(n)=2^3=8 so there are 8 regular hexagons in total.
If 2 points are drawn on each side then n=2+1=3 and a(n)=3^3=27 so
there are 27 regular hexagons in total. - Noah Priluck (npriluck(AT)gmail.com),
May 02 2007
%C A000578 a(n) = {least common multiple of n and (n-1)^2}-(n-1)^2. E.g.: {least
common multiple of 1 and (1-1)^2}-(1-1)^2 = 0, {least common multiple
of 2 and (2-1)^2}-(2-1)^2 = 1, {least common multiple of 3 and (3-1)^2}-(3-1)^2
= 8, ... - Mats Granvik (mgranvik(AT)abo.fi), Sep 24 2007
%C A000578 The solutions of the Diophantine equation: (X/Y)^2 - XY = 0 are of the
form: (n^3, n) with n>=1. The solutions of the Diophantine equation:
(m^2)*(X/Y)^2k - XY = 0 are of the form: (m*n^(2k+1), m*n^(2k-1))
with m>=1, k>=1 and n>=1. The solutions of the Diophantine equation:
(m^2)*(X/Y)^(2k+1) - XY = 0 are of the form: (m*n^(k+1), m*n^k) with
m>=1, k>=1 and n>=1. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Oct
04 2007
%C A000578 Excepting for the first two terms, the sequence corresponds to the Wiener
indices of C_{2n} i.e., the cycle on 2n vertices (n > 1). [From Kailasam
Viswanathan Iyer (kvi(AT)nitt.edu), Mar 16 2009]
%C A000578 Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 7, 4,
5, 6, 3, 2, 9. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep
04 2009]
%C A000578 a(n) = A007531(n) + A000567(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Sep 18 2009]
%C A000578 Totally multiplicative sequence with a(p) = p^3 for prime p. [From Jaroslav
Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009]
%D A000578 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000578 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000578 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 A000578 T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal,
4 (2003), 439-445.
%D A000578 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq.
(8).
%D A000578 D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.
%H A000578 N. J. A. Sloane, <a href="b000578.txt">Table of n, a(n) for n = 0..10000</
a>
%H A000578 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000578 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas
for Some Functions on Finite Sets</a>
%H A000578 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 A000578 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 A000578 H. Bottomley, <a href="a578.gif">Illustration of initial terms</a>
%H A000578 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">
Some Smarandache-type multiplicative sequences</a>
%H A000578 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A000578 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CubicNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000578 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
HexPyramidalNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000578 Ronald Yannone, <a href="http://www.polymath-systems.com/intel/hiqsocs/
megasoc/noes149/hilbert.html">Hilbert Matrix Analyses</a>
%F A000578 Multiplicative with a(p^e) = p^(3e). - David W. Wilson (davidwwilson(AT)comcast.net),
Aug 01, 2001.
%F A000578 G.f.: x(1+4x+x^2)/(1-x)^4. - Michael Somos, May 06 2003
%F A000578 Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters,
Sep 11 2005. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep
09 2005
%F A000578 E.g.f.: (x+3x^2+x^3)*e^x. - Franklin T. Adams-Watters, Sep 11 2005. -
Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
%F A000578 a(n)=sum(sum(n, j=1..n),k=1..n), n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 11 2007
%F A000578 a(n) = Sum(Sum(A002024(j,i): i<=j<n+i): 1<=i<=n). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Jun 24 2007
%F A000578 Starting (1, 8, 27, 64, 125,...), = binomial tansform of [1, 7, 12, 6,
0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007
%F A000578 a(n) = C(n+2,3) + 4 C(n+1,3) + C(n,3)
%F A000578 This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*
...* (n+k) *(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=0. - Alexander
R. Povolotsky (pevnev(AT)juno.com), May 17 2008
%F A000578 G.f.: sage: taylor( mul( x*(x^2+4*x+1)/(x-1)^4 for i in xrange(1,2)),
x,0,40)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun
03 2009]
%p A000578 A000578 := n->n^3;
%p A000578 a:=n->sum(sum(n, j=1..n),k=1..n): seq(a(n), n=0..40); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), May 11 2007
%p A000578 with(combinat):a:=n->sum(sum(sum(binomial(5,2)/10, j=0..n), k=0..n),m=0..n):
seq(a(n), n=-1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 30 2007
%p A000578 A000578:=(1+4*z+z**2)/(z-1)^4; [S. Plouffe in his 1992 dissertation if
sequence starts at a(1).]
%t A000578 Table[n^3, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com),
Apr 01 2006
%o A000578 (PARI) A000578(n)=n^3 - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 12
2008
%o A000578 isA000578(n)={n==round(sqrtn(n,3))^3} - M. F. Hasler (www.univ-ag.fr/
~mhasler), Apr 12 2008
%Y A000578 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003,
A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467,
A007588, A062025, A063521, A063522, A063523.
%Y A000578 Cf. A065876.
%Y A000578 a(n)= sum (A003215)
%Y A000578 Cf. A030078(n)=A000578(A000040(n)): cubes of primes; sums of cubes: A003325,
A024670 and references therein: A003072, ...
%Y A000578 Cf. A101102, A101097, A101094, A024166, A000537.
%Y A000578 Subsequence of A145784. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 19 2008]
%Y A000578 Sequence in context: A069939 A118880 A048390 this_sequence A062292 A030295
A052045
%Y A000578 Adjacent sequences: A000575 A000576 A000577 this_sequence A000579 A000580
A000581
%K A000578 nonn,core,easy,nice,mult,new
%O A000578 0,3
%A A000578 N. J. A. Sloane (njas(AT)research.att.com).
%E A000578 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000
%E A000578 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
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