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A000578
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The cubes: a(n) = n^3.
(Formerly M4499 N1905)
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276
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0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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
n^3 is the sum of the first n centered hexagonal numbers (A003215). - Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 29 2004
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 07 2004.
Schlaefli symbol for this polyhedron: {4,3}
Least multiple of n such that every partial sum is a square. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
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
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
The solutions of the Diophantine equation: (X/Y)^2 - X*Y = 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
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]
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]
a(n) = A007531(n) + A000567(n). [From Reinhard Zumkeller, Sep 18 2009]
Totally multiplicative sequence with a(p) = p^3 for prime p. [From Jaroslav Krizek, Nov 01 2009]
Sums of rows of the triangle in A176271, n>0. [From Reinhard Zumkeller, Apr 13 2010]
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). [From Daniel Forgues, May 14 2010]
Numbers n for which order of torsion subgroup t of the elliptic curve y^2=x^3-n is t=2. [From Artur Jasinski (grafix(AT)csl.pl), Jun 30 2010]
The sequence with the lengths of the Pisano periods mod k is 1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20,.. for k>=1, apparently multiplicative and derived from A000027 by dividing every ninth term through 3. Cubic variant of A186646. - R. J. Mathar, Mar 10 2011
The number of atoms in a bcc (body-centered cubic) rhombic hexahedron with n shells is n^3 (T.P. Martin, Shells of atoms, eq.(8)). - Brigitte Stepanov, Jul 02 2011
A010057(a(n)) = 1. [Reinhard Zumkeller, Oct 22 2011]
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REFERENCES
| T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (8).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..10000
H. Bottomley, Illustration of initial terms
Ralph Greenberg, Math For Poets
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Hyun Kwang Kim, On Regular Polytope Numbers
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.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Hex Pyramidal Number
Ronald Yannone, Hilbert Matrix Analyses
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| Multiplicative with a(p^e) = p^(3e). - David W. Wilson, Aug 01, 2001.
G.f.: x*(1+4*x+x^2)/(1-x)^4. - Michael Somos, May 06 2003
Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters, Sep 11 2005. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
E.g.f.: (x+3*x^2+x^3)*exp(x). - Franklin T. Adams-Watters, Sep 11 2005. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 09 2005
a(n) = Sum(Sum(A002024(j,i): i<=j<n+i): 1<=i<=n). - Reinhard Zumkeller, Jun 24 2007
Starting (1, 8, 27, 64, 125,...), = binomial tansform of [1, 7, 12, 6, 0, 0, 0,...]. - Gary W. Adamson, Nov 21 2007
a(n) = C(n+2,3) + 4 C(n+1,3) + C(n,3)
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, May 17 2008
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MAPLE
| A000578 := n->n^3;
A000578:=(1+4*z+z**2)/(z-1)^4; [S. Plouffe in his 1992 dissertation if sequence starts at a(1).]
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MATHEMATICA
| Table[n^3, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
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PROG
| (PARI) A000578(n)=n^3 - M. F. Hasler, Apr 12 2008
(PARI) isA000578(n)={n==round(sqrtn(n, 3))^3} - M. F. Hasler, Apr 12 2008
(Haskell)
a000578 n = a000578_list !! n
a000578_list = map (^ 3) [0..] -- Reinhard Zumkeller, Oct 22 2011
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CROSSREFS
| 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.
Cf. A065876, A101102, A101097, A101094, A024166, A000537.
a(n)= sum (A003215)
Cf. A030078(n)=A000578(A000040(n)): cubes of primes; sums of cubes: A003325, A024670 and references therein: A003072, ...
Subsequence of A145784.
Cf. A007412 (complement), A048766.
Sequence in context: A069939 A118880 A048390 * A062292 A030295 A052045
Adjacent sequences: A000575 A000576 A000577 * A000579 A000580 A000581
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KEYWORD
| nonn,core,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000
Removed attribute "conjectured" from Plouffe g.f., R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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