|
%I M4499 N1905 #491 Feb 28 2024 06:32:26
%S 0,1,8,27,64,125,216,343,512,729,1000,1331,1728,2197,2744,3375,4096,
%T 4913,5832,6859,8000,9261,10648,12167,13824,15625,17576,19683,21952,
%U 24389,27000,29791,32768,35937,39304,42875,46656,50653,54872,59319,64000
%N The cubes: a(n) = n^3.
%C 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 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
%C 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 Schlaefli symbol for this polyhedron: {4, 3}.
%C Least multiple of n such that every partial sum is a square. - _Amarnath Murthy_, Sep 09 2005
%C 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 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 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 Totally multiplicative sequence with a(p) = p^3 for prime p. - _Jaroslav Krizek_, Nov 01 2009
%C Sums of rows of the triangle in A176271, n > 0. - _Reinhard Zumkeller_, Apr 13 2010
%C One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - _Daniel Forgues_, May 14 2010
%C 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 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 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 The inverse binomial transform yields the (finite) 0, 1, 6, 6 (third row in A019538 and A131689). - _R. J. Mathar_, Jan 16 2013
%C 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 If n > 0 is not congruent to 5 (mod 6) then A010888(a(n)) divides a(n). - _Ivan N. Ianakiev_, Oct 16 2013
%C 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 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 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 From _Bui Quang Tuan_, Mar 31 2015: (Start)
%C 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 Here is the example for n = 4, where 1 + 2*2 + 3*3 + 4*4 + 3*5 + 2*6 + 7 = 64 = a(4):
%C 1
%C 2 2
%C 3 3 3
%C 4 4 4 4
%C 5 5 5
%C 6 6
%C 7
%C (End)
%C 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 Does not satisfy Benford's law [Ross, 2012]. - _N. J. A. Sloane_, Feb 08 2017
%C 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 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 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 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 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.
%H N. J. A. Sloane, <a href="/A000578/b000578.txt">Table of n, a(n) for n = 0..10000</a>
%H H. Bottomley, <a href="/A000578/a000578.gif">Illustration of initial terms</a>
%H British National Museum, <a href="https://www.britishmuseum.org/collection/object/W_-92698">Tablet 92698</a>
%H 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 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 Ralph Greenberg, <a href="http://www.math.washington.edu/~greenber/MathPoet.html">Math For Poets</a>
%H 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 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 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 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 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 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 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>, and <a href="http://mathworld.wolfram.com/HexPyramidalNumber.html">Hex Pyramidal Number</a>
%H Ronald Yannone, <a href="http://megasociety.org/noesis/149/hilbert.html">Hilbert Matrix Analyses</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = Sum_{i=0..n-1} A003215(i).
%F Multiplicative with a(p^e) = p^(3e). - _David W. Wilson_, Aug 01 2001
%F G.f.: x*(1+4*x+x^2)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation
%F Dirichlet generating function: zeta(s-3). - _Franklin T. Adams-Watters_, Sep 11 2005, _Amarnath Murthy_, Sep 09 2005
%F E.g.f.: (1+3*x+x^2)*x*exp(x). - _Franklin T. Adams-Watters_, Sep 11 2005 - _Amarnath Murthy_, Sep 09 2005
%F a(n) = Sum_{i=1..n} (Sum_{j=i..n+i-1} A002024(j,i)). - _Reinhard Zumkeller_, Jun 24 2007
%F 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 Starting (1, 8, 27, 64, 125, ...), = binomial transform of [1, 7, 12, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 21 2007
%F a(n) = A007531(n) + A000567(n). - _Reinhard Zumkeller_, Sep 18 2009
%F 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 a(n) = n + 6*binomial(n+1,3) = binomial(n,1)+6*binomial(n+1,3). - _Ron Knott_, Jun 10 2019
%F A010057(a(n)) = 1. - _Reinhard Zumkeller_, Oct 22 2011
%F a(n) = A000537(n) - A000537(n-1), difference between 2 squares of consecutive triangular numbers. - _Pierre CAMI_, Feb 20 2012
%F a(n) = A048395(n) - 2*A006002(n). - _J. M. Bergot_, Nov 25 2012
%F a(n) = 1 + 7*(n-1) + 6*(n-1)*(n-2) + (n-1)*(n-2)*(n-3). - _Antonio Alberto Olivares_, Apr 03 2013
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6. - _Ant King_ Apr 29 2013
%F a(n) = A000330(n) + Sum_{i=1..n-1} A014105(i), n >= 1. - _Ivan N. Ianakiev_, Sep 20 2013
%F 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 For n >= 1, a(n) = A152618(n-1) + A033996(n-1). - _Bui Quang Tuan_, Apr 01 2015
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Jon Tavasanis_, Feb 21 2016
%F 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 a(n) = A000292(n-1) * 6 + n. - _Zhandos Mambetaliyev_, Nov 24 2016
%F a(n) = n*binomial(n+1, 2) + 2*binomial(n+1, 3) + binomial(n,3). - _Tony Foster III_, Nov 14 2017
%F From _Amiram Eldar_, Jul 02 2020: (Start)
%F Sum_{n>=1} 1/a(n) = zeta(3) (A002117).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/4 (A197070). (End)
%F From _Amiram Eldar_, Jan 20 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi.
%F Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)/(3*Pi). (End)
%F 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 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 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 := n->n^3;
%p seq(A000578(n), n=0..50);
%p isA000578 := proc(r)
%p local p;
%p if r = 0 or r =1 then
%p true;
%p else
%p for p in ifactors(r)[2] do
%p if op(2, p) mod 3 <> 0 then
%p return false;
%p end if;
%p end do:
%p true ;
%p end if;
%p end proc: # _R. J. Mathar_, Oct 08 2013
%t Table[n^3, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Jul 05 2014 *)
%t 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 (PARI) A000578(n)=n^3 \\ _M. F. Hasler_, Apr 12 2008
%o (PARI) is(n)=ispower(n,3) \\ _Charles R Greathouse IV_, Feb 20 2012
%o (Haskell)
%o a000578 = (^ 3)
%o a000578_list = 0 : 1 : 8 : zipWith (+)
%o (map (+ 6) a000578_list)
%o (map (* 3) $ tail $ zipWith (-) (tail a000578_list) a000578_list)
%o -- _Reinhard Zumkeller_, Sep 05 2015, May 24 2012, Oct 22 2011
%o (Maxima) A000578(n):=n^3$
%o makelist(A000578(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%o (Magma) [ n^3 : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 14 2014
%o (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 (Python)
%o A000578_list, m = [], [6, -6, 1, 0]
%o for _ in range(10**2):
%o A000578_list.append(m[-1])
%o for i in range(3):
%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o (Scheme) (define (A000578 n) (* n n n)) ;; _Antti Karttunen_, Oct 06 2017
%Y (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 For sums of cubes, cf. A000537 (partial sums), A003072, A003325, A024166, A024670, A101102 (fifth partial sums).
%Y Cf. A001158 (inverse Möbius transform), A007412 (complement), A030078(n) (cubes of primes), A048766, A058645 (binomial transform), A065876, A101094, A101097.
%Y Subsequence of A145784.
%Y Cf. A260260 (comment). - _Bruno Berselli_, Jul 22 2015
%Y Cf. A000292 (tetrahedral numbers), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
%Y Cf. A098737 (main diagonal).
%K nonn,core,easy,nice,mult
%O 0,3
%A _N. J. A. Sloane_
|
