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A000578 The cubes: a(n) = n^3.

%I A000578 M4499 N1905 #521 Jun 27 2025 15:34:49
%S A000578 0,1,8,27,64,125,216,343,512,729,1000,1331,1728,2197,2744,3375,4096,
%T A000578 4913,5832,6859,8000,9261,10648,12167,13824,15625,17576,19683,21952,
%U A000578 24389,27000,29791,32768,35937,39304,42875,46656,50653,54872,59319,64000,68921,74088,79507
%N A000578 The cubes: a(n) = n^3.
%C A000578 a(n) is the 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
%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_, Jun 02 2004. See Propp and Propp-Gubin for a proof.
%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 07 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_, 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 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_, Oct 04 2007
%C A000578 Except 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
%C A000578 Totally multiplicative sequence with a(p) = p^3 for prime p. - _Jaroslav Krizek_, Nov 01 2009
%C A000578 Sums of rows of the triangle in A176271, n > 0. - _Reinhard Zumkeller_, Apr 13 2010
%C A000578 One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - _Daniel Forgues_, May 14 2010
%C A000578 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
%C A000578 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
%C A000578 The number of atoms in a bcc (body-centered cubic) rhombic hexahedron with n atoms along one edge is n^3 (T. P. Martin, Shells of atoms, eq. (8)). - _Brigitte Stepanov_, Jul 02 2011
%C A000578 The inverse binomial transform yields the (finite) 0, 1, 6, 6 (third row in A019538 and A131689). - _R. J. Mathar_, Jan 16 2013
%C A000578 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 triangular numbers. - _J. M. Bergot_, Jun 25 2013
%C A000578 If n > 0 is not congruent to 5 (mod 6) then A010888(a(n)) divides a(n). - _Ivan N. Ianakiev_, Oct 16 2013
%C A000578 For n > 2, a(n) = twice the area of a triangle with vertices at points (binomial(n,3),binomial(n+2,3)), (binomial(n+1,3),binomial(n+1,3)), and (binomial(n+2,3),binomial(n,3)). - _J. M. Bergot_, Jun 14 2014
%C A000578 Determinants of the spiral knots S(4,k,(1,1,-1)). a(k) = det(S(4,k,(1,1,-1))). - _Ryan Stees_, Dec 14 2014
%C A000578 One of the oldest-known examples of this sequence is shown in the Senkereh tablet, BM 92698, which displays the first 32 terms in cuneiform. - _Charles R Greathouse IV_, Jan 21 2015
%C A000578 From _Bui Quang Tuan_, Mar 31 2015: (Start)
%C A000578 We construct a number triangle from the integers 1, 2, 3, ... 2*n-1 as follows. The first column contains all the integers 1, 2, 3, ... 2*n-1. Each succeeding column is the same as the previous column but without the first and last items. The last column contains only n. The sum of all the numbers in the triangle is n^3.
%C A000578 Here is the example for n = 4, where 1 + 2*2 + 3*3 + 4*4 + 3*5 + 2*6 + 7 = 64 = a(4):
%C A000578   1
%C A000578   2  2
%C A000578   3  3  3
%C A000578   4  4  4  4
%C A000578   5  5  5
%C A000578   6  6
%C A000578   7
%C A000578 (End)
%C A000578 For n > 0, a(n) is the number of compositions of n+11 into n parts avoiding parts 2 and 3. - _Milan Janjic_, Jan 07 2016
%C A000578 Does not satisfy Benford's law [Ross, 2012]. - _N. J. A. Sloane_, Feb 08 2017
%C A000578 Number of inequivalent face colorings of the cube using at most n colors such that each color appears at least twice. - _David Nacin_, Feb 22 2017
%C A000578 Consider A = {a,b,c} a set with three distinct members. The number of subsets of A is 8, including {a,b,c} and the empty set. The number of subsets from each of those 8 subsets is 27. If the number of such iterations is n, then the total number of subsets is a(n-1). - _Gregory L. Simay_, Jul 27 2018
%C A000578 By Fermat's Last Theorem, these are the integers of the form x^k with the least possible value of k such that x^k = y^k + z^k never has a solution in positive integers x, y, z for that k. - _Felix Fröhlich_, Jul 27 2018
%D A000578 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 191.
%D A000578 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 43, 64, 81.
%D A000578 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
%D A000578 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 292.
%D A000578 T. Aaron Gulliver, "Sequences from cubes of integers", International Mathematical Journal, 4 (2003), no. 5, 439 - 445. See http://www.m-hikari.com/z2003.html for information about this journal. [I expanded the reference to make this easier to find. - _N. J. A. Sloane_, Feb 18 2019]
%D A000578 J. Propp and A. Propp-Gubin, "Counting Triangles in Triangles", Pi Mu Epsilon Journal (to appear).
%D A000578 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000578 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000578 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 6-7.
%D A000578 D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.
%H A000578 N. J. A. Sloane, <a href="/A000578/b000578.txt">Table of n, a(n) for n = 0..10000</a>
%H A000578 H. Bottomley, <a href="/A000578/a000578.gif">Illustration of initial terms</a>
%H A000578 British National Museum, <a href="https://www.britishmuseum.org/collection/object/W_-92698">Tablet 92698</a>
%H A000578 N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, <a href="http://projecteuclid.org/euclid.mjms/1312232716">Spiral knots</a>, Missouri J. of Math. Sci., 22 (2010).
%H A000578 M. DeLong, M. Russell, and J. Schrock, <a href="http://dx.doi.org/10.2140/involve.2015.8.361">Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)</a>, Involve, Vol. 8 (2015), No. 3, 361-384.
%H A000578 Ralph Greenberg, <a href="http://www.math.washington.edu/~greenber/MathPoet.html">Math For Poets</a>
%H A000578 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A000578 Milan Janjic, <a href="https://web.archive.org/web/20150919034515/http://www.pmfbl.org/janjic/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a> [Cached version at the Wayback Machine]
%H A000578 Hyun Kwang Kim, <a href="https://doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75. - fixed by _Felix Fröhlich_, Jun 16 2014
%H A000578 T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (8).
%H A000578 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H A000578 Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
%H A000578 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000578 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000578 James Propp and Adam Propp-Gubin, <a href="https://arxiv.org/abs/2409.17117">Counting Triangles in Triangles</a>, arXiv:2409.17117 [math.CO], 25 September 2024.
%H A000578 Kenneth A. Ross, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.036">First Digits of Squares and Cubes</a>, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
%H A000578 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>, and <a href="https://mathworld.wolfram.com/HexPyramidalNumber.html">Hex Pyramidal Number</a>
%H A000578 Ronald Yannone, <a href="http://megasociety.org/noesis/149/hilbert.html">Hilbert Matrix Analyses</a>
%H A000578 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%H A000578 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000578 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000578 a(n) = Sum_{i=0..n-1} A003215(i).
%F A000578 Multiplicative with a(p^e) = p^(3e). - _David W. Wilson_, Aug 01 2001
%F A000578 G.f.: x*(1+4*x+x^2)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation
%F A000578 Dirichlet generating function: zeta(s-3). - _Franklin T. Adams-Watters_, Sep 11 2005, _Amarnath Murthy_, Sep 09 2005
%F A000578 E.g.f.: (1+3*x+x^2)*x*exp(x). - _Franklin T. Adams-Watters_, Sep 11 2005 - _Amarnath Murthy_, Sep 09 2005
%F A000578 a(n) = Sum_{i=1..n} (Sum_{j=i..n+i-1} A002024(j,i)). - _Reinhard Zumkeller_, Jun 24 2007
%F A000578 a(n) = lcm(n, (n - 1)^2) - (n - 1)^2. E.g.: lcm(1, (1 - 1)^2) - (1 - 1)^2 = 0, lcm(2, (2 - 1)^2) - (2 - 1)^2 = 1, lcm(3, (3 - 1)^2) - (3 - 1)^2 = 8, ... - _Mats Granvik_, Sep 24 2007
%F A000578 Starting (1, 8, 27, 64, 125, ...), = binomial transform of [1, 7, 12, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 21 2007
%F A000578 a(n) = A007531(n) + A000567(n). - _Reinhard Zumkeller_, Sep 18 2009
%F A000578 a(n) = binomial(n+2,3) + 4*binomial(n+1,3) + binomial(n,3). [Worpitzky's identity for cubes. See. e.g., Graham et al., eq. (6.37). - _Wolfdieter Lang_, Jul 17 2019]
%F A000578 a(n) = n + 6*binomial(n+1,3) = binomial(n,1)+6*binomial(n+1,3). - _Ron Knott_, Jun 10 2019
%F A000578 A010057(a(n)) = 1. - _Reinhard Zumkeller_, Oct 22 2011
%F A000578 a(n) = A000537(n) - A000537(n-1), difference between 2 squares of consecutive triangular numbers. - _Pierre CAMI_, Feb 20 2012
%F A000578 a(n) = A048395(n) - 2*A006002(n). - _J. M. Bergot_, Nov 25 2012
%F A000578 a(n) = 1 + 7*(n-1) + 6*(n-1)*(n-2) + (n-1)*(n-2)*(n-3). - _Antonio Alberto Olivares_, Apr 03 2013
%F A000578 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6. - _Ant King_ Apr 29 2013
%F A000578 a(n) = A000330(n) + Sum_{i=1..n-1} A014105(i), n >= 1. - _Ivan N. Ianakiev_, Sep 20 2013
%F A000578 a(k) = det(S(4,k,(1,1,-1))) = k*b(k)^2, where b(1)=1, b(2)=2, b(k) = 2*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - _Ryan Stees_, Dec 14 2014
%F A000578 For n >= 1, a(n) = A152618(n-1) + A033996(n-1). - _Bui Quang Tuan_, Apr 01 2015
%F A000578 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Jon Tavasanis_, Feb 21 2016
%F A000578 a(n) = n + Sum_{j=0..n-1} Sum_{k=1..2} binomial(3,k)*j^(3-k). - _Patrick J. McNab_, Mar 28 2016
%F A000578 a(n) = A000292(n-1) * 6 + n. - _Zhandos Mambetaliyev_, Nov 24 2016
%F A000578 a(n) = n*binomial(n+1, 2) + 2*binomial(n+1, 3) + binomial(n,3). - _Tony Foster III_, Nov 14 2017
%F A000578 From _Amiram Eldar_, Jul 02 2020: (Start)
%F A000578 Sum_{n>=1} 1/a(n) = zeta(3) (A002117).
%F A000578 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/4 (A197070). (End)
%F A000578 From _Amiram Eldar_, Jan 20 2021: (Start)
%F A000578 Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi.
%F A000578 Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)/(3*Pi). (End)
%F A000578 a(n) = Sum_{d|n} sigma_3(d)*mu(n/d) = Sum_{d|n} A001158(d)*A008683(n/d). Moebius transform of sigma_3(n). - _Ridouane Oudra_, Apr 15 2021
%e A000578 For k=3, b(3) = 2 b(2) - b(1) = 4-1 = 3, so det(S(4,3,(1,1,-1))) = 3*3^2 = 27.
%e A000578 For n=3, a(3) = 3 + (3*0^2 + 3*0 + 3*1^2 + 3*1 + 3*2^2 + 3*2) = 27. - _Patrick J. McNab_, Mar 28 2016
%p A000578 A000578 := n->n^3;
%p A000578 seq(A000578(n), n=0..50);
%p A000578 isA000578 := proc(r)
%p A000578     local p;
%p A000578     if r = 0 or r =1 then
%p A000578         true;
%p A000578     else
%p A000578         for p in ifactors(r)[2] do
%p A000578             if op(2, p) mod 3 <> 0 then
%p A000578                 return false;
%p A000578             end if;
%p A000578         end do:
%p A000578         true ;
%p A000578     end if;
%p A000578 end proc: # _R. J. Mathar_, Oct 08 2013
%t A000578 Table[n^3, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t A000578 CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Jul 05 2014 *)
%t A000578 Accumulate[Table[3n^2+3n+1,{n,0,20}]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,27,64},20](* _Harvey P. Dale_, Aug 18 2018 *)
%o A000578 (PARI) A000578(n)=n^3 \\ _M. F. Hasler_, Apr 12 2008
%o A000578 (PARI) is(n)=ispower(n,3) \\ _Charles R Greathouse IV_, Feb 20 2012
%o A000578 (Haskell)
%o A000578 a000578 = (^ 3)
%o A000578 a000578_list = 0 : 1 : 8 : zipWith (+)
%o A000578    (map (+ 6) a000578_list)
%o A000578    (map (* 3) $ tail $ zipWith (-) (tail a000578_list) a000578_list)
%o A000578 -- _Reinhard Zumkeller_, Sep 05 2015, May 24 2012, Oct 22 2011
%o A000578 (Maxima) A000578(n):=n^3$
%o A000578 makelist(A000578(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%o A000578 (Magma) [ n^3 : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 14 2014
%o A000578 (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
%o A000578 (Python)
%o A000578 A000578_list, m = [], [6, -6, 1, 0]
%o A000578 for _ in range(10**2):
%o A000578     A000578_list.append(m[-1])
%o A000578     for i in range(3):
%o A000578         m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o A000578 (Scheme) (define (A000578 n) (* n n n)) ;; _Antti Karttunen_, Oct 06 2017
%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 For sums of cubes, cf. A000537 (partial sums), A003072, A003325, A024166, A024670, A101102 (fifth partial sums).
%Y A000578 Cf. A001158 (inverse Möbius transform), A007412 (complement), A030078(n) (cubes of primes), A048766, A058645 (binomial transform), A065876, A101094, A101097.
%Y A000578 Subsequence of A145784.
%Y A000578 Cf. A260260 (comment). - _Bruno Berselli_, Jul 22 2015
%Y A000578 Cf. A000292 (tetrahedral numbers), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
%Y A000578 Cf. A098737 (main diagonal).
%K A000578 nonn,core,easy,nice,mult
%O A000578 0,3
%A A000578 _N. J. A. Sloane_