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A000578 The cubes: a(n) = n^3.
(Formerly M4499 N1905)
410
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; text; internal format)
OFFSET

0,3

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, 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, Jun 02 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, 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, 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). - K.V.Iyer, Mar 16 2009

Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 7, 4, 5, 6, 3, 2, 9. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009

a(n) = A007531(n) + A000567(n). - Reinhard Zumkeller, Sep 18 2009

Totally multiplicative sequence with a(p) = p^3 for prime p. - Jaroslav Krizek, Nov 01 2009

Sums of rows of the triangle in A176271, n > 0. - Reinhard Zumkeller, Apr 13 2010

One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - 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. - Artur Jasinski, 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

The inverse binomial transform yields the (finite) 0, 1, 6, 6 (third row in A019538 and A131689). - R. J. Mathar, Jan 16 2013

Twice the area of a triangle with vertices at (0, 0), (t(n - 1), t(n)), and (t(n), t(n - 1)), where t = A000217 are triangle numbers. - J. M. Bergot, Jun 25 2013

If n > 0 is not congruent to 5 (mod 6) then A010888(a(n)) divides a(n). - Ivan N. Ianakiev, Oct 16 2013

For n>2, a(n)=twice the area of a triangle with vertices at points (C(n,3),C(n+2,3)), (C(n+1,3),C(n+1,3)), and (C(n+2,3),C(n,3). - J. M. Bergot, Jun 14 2014

27, 64, 343, and 1331 are conjectured to be the only cubes not divisible by 10 with 2 distinct digits. See A155146 for cubes with 3 distinct digits and A155147 for cubes with 4 distinct digits. - Derek Orr, Sep 23 2014

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.

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, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Cubic Number

Eric Weisstein's World of Mathematics, Hex Pyramidal Number

Ronald Yannone, Hilbert Matrix Analyses

Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = sum_{i=0..n-1} A003215(i).

Multiplicative with a(p^e) = p^(3e). - David W. Wilson, Aug 01 2001

G.f.: x*(1+4*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation

Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters, Sep 11 2005, Amarnath Murthy, Sep 09 2005

E.g.f.: (x+3*x^2+x^3)*exp(x). - Franklin T. Adams-Watters, Sep 11 2005 - Amarnath Murthy, 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 transform 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

a(n) = A000537(n) - A000537(n-1), difference between 2 squares of consecutive triangular numbers. - Pierre CAMI, Feb 20 2012

a(n) = A048395(n) - 2*A006002(n). - J. M. Bergot, Nov 25 2012

a(n) = 1 + 7(n-1) + 6(n-1)(n-2) + (n-1)(n-2)(n-3). - Antonio Alberto Olivares, Apr 03 2013

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+6. - Ant King Apr 29 2013

a(n) = A000330(n) + sum(i=1..n-1, A014105(i)), n>=1. - Ivan N. Ianakiev, Sep 20 2013

MAPLE

A000578 := n->n^3; seq(A000578(n), n=0..50);

isA00578 := proc(r)

    local p;

    if r = 0 or r =1 then

        true;

    else

        for p in ifactors(r)[2] do

            if op(2, p) mod 3 <> 0 then

                return false;

            end if;

        end do:

        true ;

    end if;

end proc: # R. J. Mathar, Oct 08 2013

MATHEMATICA

Table[n^3, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)

CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jul 05 2014 *)

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

(PARI) is(n)=ispower(n, 3) \\ Charles R Greathouse IV, Feb 20 2012

(Haskell)

a000578 = (^ 3)

a000578_list = 0 : map sum a176271_tabl

-- Reinhard Zumkeller, May 24 2012, Oct 22 2011

(Maxima) A000578(n):=n^3$

makelist(A000578(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

(MAGMA) [ n^3 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

(MAGMA) I:=[0, 1, 8, 27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jul 05 2014

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.

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, A058645 (binomial transform).

Sequence in context: A069939 A118880 A048390 * A062292 A030295 A052045

Adjacent sequences:  A000575 A000576 A000577 * A000579 A000580 A000581

KEYWORD

nonn,core,easy,nice,mult

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

More terms from James A. Sellers, Jun 20 2000

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

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Last modified November 27 01:09 EST 2014. Contains 250152 sequences.