A000330 - OEIS

%I M3844 N1574 #625 Dec 07 2024 08:02:18

%S 0,1,5,14,30,55,91,140,204,285,385,506,650,819,1015,1240,1496,1785,

%T 2109,2470,2870,3311,3795,4324,4900,5525,6201,6930,7714,8555,9455,

%U 10416,11440,12529,13685,14910,16206,17575,19019,20540,22140,23821,25585,27434,29370

%N Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.

%C The sequence contains exactly one square greater than 1, namely 4900 (according to Gardner). - _Jud McCranie_, Mar 19 2001, Mar 22 2007 [This is a result from Watson. - _Charles R Greathouse IV_, Jun 21 2013] [See A351830 for further related comments and references.]

%C Number of rhombi in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

%C Number of acute triangles made from the vertices of a regular n-polygon when n is odd (cf. A007290). - _Sen-Peng Eu_, Apr 05 2001

%C Gives number of squares formed from an n X n square. In a 1 X 1 square, one is formed. In a 2 X 2 square, five squares are formed. In a 3 X 3 square, 14 squares are formed and so on. - Kristie Smith (kristie10spud(AT)hotmail.com), Apr 16 2002

%C a(n-1) = B_3(n)/3, where B_3(x) = x(x-1)(x-1/2) is the third Bernoulli polynomial. - _Michael Somos_, Mar 13 2004

%C Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly once.

%C Since 3*r = (r+1) + r + (r-1) = T(r+1) - T(r-2), where T(r) = r-th triangular number r*(r+1)/2, we have 3*r^2 = r*(T(r+1) - T(r-2)) = f(r+1) - f(r-1) ... (i), where f(r) = (r-1)*T(r) = (r+1)*T(r-1). Summing over n, the right hand side of relation (i) telescopes to f(n+1) + f(n) = T(n)*((n+2) + (n-1)), whence the result Sum_{r=1..n} r^2 = n*(n+1)*(2*n+1)/6 immediately follows. - _Lekraj Beedassy_, Aug 06 2004

%C Also as a(n) = (1/6)*(2*n^3 + 3*n^2 + n), n > 0: structured trigonal diamond numbers (vertex structure 5) (cf. A006003 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

%C Number of triples of integers from {1, 2, ..., n} whose last component is greater than or equal to the others.

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 12 2005

%C Sum of the first n positive squares. - _Cino Hilliard_, Jun 18 2007

%C Maximal number of cubes of side 1 in a right pyramid with a square base of side n and height n. - Pasquale CUTOLO (p.cutolo(AT)inwind.it), Jul 09 2007

%C If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 4-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 19 2007

%C We also have the identity 1 + (1+4) + (1+4+9) + ... + (1+4+9+16+ ... + n^2) = n(n+1)(n+2)(n+(n+1)+(n+2))/36; ... and in general the k-fold nested sum of squares can be expressed as n(n+1)...(n+k)(n+(n+1)+...+(n+k))/((k+2)!(k+1)/2). - _Alexander R. Povolotsky_, Nov 21 2007

%C The terms of this sequence are coefficients of the Engel expansion of the following converging sum: 1/(1^2) + (1/1^2)*(1/(1^2+2^2)) + (1/1^2)*(1/(1^2+2^2))*(1/(1^2+2^2+3^2)) + ... - _Alexander R. Povolotsky_, Dec 10 2007

%C Convolution of A000290 with A000012. - _Sergio Falcon_, Feb 05 2008

%C Hankel transform of binomial(2*n-3, n-1) is -a(n). - _Paul Barry_, Feb 12 2008

%C Starting (1, 5, 14, 30, ...) = binomial transform of [1, 4, 5, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 13 2008

%C Starting (1,5,14,30,...) = second partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+2,i+2)*b(i), where b(i)=1,2,0,0,0,... - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

%C Convolution of A001477 with A005408: a(n) = Sum_{k=0..n} (2*k+1)*(n-k). - _Reinhard Zumkeller_, Mar 07 2009

%C Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF1 denominators of A156921. See A157702 for background information. - _Johannes W. Meijer_, Mar 07 2009

%C The sequence is related to A000217 by a(n) = n*A000217(n) - Sum_{i=0..n-1} A000217(i) and this is the case d = 1 in the identity n^2*(d*n-d+2)/2 - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)(2*d*n-2*d+3)/6, or also the case d = 0 in n^2*(n+2*d+1)/2 - Sum_{i=0..n-1} i*(i+2*d+1)/2 = n*(n+1)*(2*n+3*d+1)/6. - _Bruno Berselli_, Apr 21 2010, Apr 03 2012

%C a(n)/n = k^2 (k = integer) for n = 337; a(337) = 12814425, a(n)/n = 38025, k = 195, i.e., the number k = 195 is the quadratic mean (root mean square) of the first 337 positive integers. There are other such numbers -- see A084231 and A084232. - _Jaroslav Krizek_, May 23 2010

%C Also the number of moves to solve the "alternate coins game": given 2n+1 coins (n+1 Black, n White) set alternately in a row (BWBW...BWB) translate (not rotate) a pair of adjacent coins at a time (1 B and 1 W) so that at the end the arrangement shall be BBBBB..BW...WWWWW (Blacks separated by Whites). Isolated coins cannot be moved. - _Carmine Suriano_, Sep 10 2010

%C From _J. M. Bergot_, Aug 23 2011: (Start)

%C Using four consecutive numbers n, n+1, n+2, and n+3 take all possible pairs (n, n+1), (n, n+2), (n, n+3), (n+1, n+2), (n+1, n+3), (n+2, n+3) to create unreduced Pythagorean triangles. The sum of all six areas is 60*a(n+1).

%C Using three consecutive odd numbers j, k, m, (j+k+m)^3 - (j^3 + k^3 + m^3) equals 576*a(n) = 24^2*a(n) where n = (j+1)/2. (End)

%C From _Ant King_, Oct 17 2012: (Start)

%C For n > 0, the digital roots of this sequence A010888(a(n)) form the purely periodic 27-cycle {1, 5, 5, 3, 1, 1, 5, 6, 6, 7, 2, 2, 9, 7, 7, 2, 3, 3, 4, 8, 8, 6, 4, 4, 8, 9, 9}.

%C For n > 0, the units' digits of this sequence A010879(a(n)) form the purely periodic 20-cycle {1, 5, 4, 0, 5, 1, 0, 4, 5, 5, 6, 0, 9, 5, 0, 6, 5, 9, 0, 0}. (End)

%C Length of the Pisano period of this sequence mod n, n>=1: 1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17, 108, 19, 40, ... . - _R. J. Mathar_, Oct 17 2012

%C Sum of entries of n X n square matrix with elements min(i,j). - _Enrique Pérez Herrero_, Jan 16 2013

%C The number of intersections of diagonals in the interior of regular n-gon for odd n > 1 divided by n is a square pyramidal number; that is, A006561(2*n+1)/(2*n+1) = A000330(n-1) = (1/6)*n*(n-1)*(2*n-1). - _Martin Renner_, Mar 06 2013

%C For n > 1, a(n)/(2n+1) = A024702(m), for n such that 2n+1 = prime, which results in 2n+1 = A000040(m). For example, for n = 8, 2n+1 = 17 = A000040(7), a(8) = 204, 204/17 = 12 = A024702(7). - _Richard R. Forberg_, Aug 20 2013

%C A formula for the r-th successive summation of k^2, for k = 1 to n, is (2*n+r)*(n+r)!/((r+2)!*(n-1)!) (H. W. Gould). - _Gary Detlefs_, Jan 02 2014

%C The n-th square pyramidal number = the n-th triangular dipyramidal number (Johnson 12), which is the sum of the n-th + (n-1)-st tetrahedral numbers. E.g., the 3rd tetrahedral number is 10 = 1+3+6, the 2nd is 4 = 1+3. In triangular "dipyramidal form" these numbers can be written as 1+3+6+3+1 = 14. For "square pyramidal form", rebracket as 1+(1+3)+(3+6) = 14. - _John F. Richardson_, Mar 27 2014

%C Beukers and Top prove that no square pyramidal number > 1 equals a tetrahedral number A000292. - _Jonathan Sondow_, Jun 21 2014

%C Odd numbered entries are related to dissections of polygons through A100157. - _Tom Copeland_, Oct 05 2014

%C From _Bui Quang Tuan_, Apr 03 2015: (Start)

%C We construct a number triangle from the integers 1, 2, 3, ..., n as follows. The first column contains 2*n-1 integers 1. The second column contains 2*n-3 integers 2, ... The last column contains only one integer n. The sum of all the numbers in the triangle is a(n).

%C Here is an example with n = 5:

%C   1

%C   1  2

%C   1  2  3

%C   1  2  3  4

%C   1  2  3  4  5

%C   1  2  3  4

%C   1  2  3

%C   1  2

%C   1

%C (End)

%C The Catalan number series A000108(n+3), offset 0, gives Hankel transform revealing the square pyramidal numbers starting at 5, A000330(n+2), offset 0 (empirical observation). - _Tony Foster III_, Sep 05 2016; see Dougherty et al. link p. 2. - _Andrey Zabolotskiy_, Oct 13 2016

%C Number of floating point additions in the factorization of an (n+1) X (n+1) real matrix by Gaussian elimination as e.g. implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of multiplications is given by A007290. - _Hugo Pfoertner_, Mar 28 2018

%C The Jacobi polynomial P(n-1,-n+2,2,3) or equivalently the sum of dot products of vectors from the first n rows of Pascal's triangle (A007318) with the up-diagonal Chebyshev T coefficient vector (1,3,2,0,...) (A053120) or down-diagonal vector (1,-7,32,-120,400,...) (A001794). a(5) = 1 + (1,1).(1,3) + (1,2,1).(1,3,2) + (1,3,3,1).(1,3,2,0) + (1,4,6,4,1).(1,3,2,0,0) = (1 + (1,1).(1,-7) + (1,2,1).(1,-7,32) + (1,3,3,1).(1,-7,32,-120) + (1,4,6,4,1).(1,-7,32,-120,400))*(-1)^(n-1) = 55. - _Richard Turk_, Jul 03 2018

%C Coefficients in the terminating series identity 1 - 5*n/(n + 4) + 14*n*(n - 1)/((n + 4)*(n + 5)) - 30*n*(n - 1)*(n - 2)/((n + 4)*(n + 5)*(n + 6)) + ... = 0 for n = 1,2,3,.... Cf. A002415 and A108674. - _Peter Bala_, Feb 12 2019

%C n divides a(n) iff n == +- 1 (mod 6) (see A007310). (See De Koninck reference.) Examples: a(11) = 506 = 11 * 46, and a(13) = 819 = 13 * 63. - _Bernard Schott_, Jan 10 2020

%C For n > 0, a(n) is the number of ternary words of length n+2 having 3 letters equal to 2 and 0 only occurring as the last letter. For example, for n=2, the length 4 words are 2221,2212,2122,1222,2220. - _Milan Janjic_, Jan 28 2020

%C Conjecture: Every integer can be represented as a sum of three generalized square pyramidal numbers. A related conjecture is given in A336205 corresponding to pentagonal case. A stronger version of these conjectures is that every integer can be expressed as a sum of three generalized r-gonal pyramidal numbers for all r >= 3. In here "generalized" means negative indices are included. - _Altug Alkan_, Jul 30 2020

%C The natural number y is a term if and only if y = a(floor((3 * y)^(1/3))). - _Robert Israel_, Dec 04 2024

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover Publications, NY, 1964, p. 194.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 215,223.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 122, see #19 (3(1)), I(n); p. 155.

%D H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).

%D J. M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 310, pp. 46-196, Ellipses, Paris, 2004.

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.

%D M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 293.

%D M. Holt, Math puzzles and games, Walker Publishing Company, 1977, p. 2 and p. 89.

%D Simon Singh, The Simpsons and Their Mathematical Secrets. London: Bloomsbury Publishing PLC (2013): 188.

%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 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 126.

%H Felix Fröhlich, <a href="/A000330/b000330.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H L. Ancora, <a href="https://drive.google.com/file/d/0B4iaQ-gBYTaJMDJFd2FFbkU2TU0/view?usp=sharing">Quadrature of the Parabola with the Square Pyramidal Number</a>, Mondadori Education, Archimede 66, No. 3, 139-144 (2014).

%H Jack Anderson, Amy Woodall, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2411.08398">Arithmetic Polygons and Sums of Consecutive Squares</a>, arXiv:2411.08398 [math.NT], 2024.

%H B. Babcock and A. van Tuyl, <a href="http://arxiv.org/abs/1109.5847">Revisiting the spreading and covering numbers</a>, arXiv preprint arXiv:1109.5847 [math.AC], 2011.

%H J. L. Bailey, Jr., <a href="http://dx.doi.org/10.1214/aoms/1177732978">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359.

%H J. L. Bailey, <a href="/A002309/a002309.pdf">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]

%H Michael A. Bennett, <a href="http://dx.doi.org/10.4064/aa105-4-3">Lucas' square pyramid problem revisited</a>, Acta Arithmetica 105 (2002), 341-347.

%H Bruno Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%H F. Beukers and J. Top, <a href="http://www.math.rug.nl/~top/oranges.pdf">On oranges and integral points on certain plane cubic curves</a>, Nieuw Arch. Wiskd., IV (1988), Ser. 6, No. 3, 203-210.

%H Henry Bottomley, <a href="/A000330/a000330.gif">Illustration of initial terms</a>.

%H S. Butler and P. Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. 13 (2010), 10.4.4, p=1 in first displayed equation page 4.

%H Bikash Chakraborty, <a href="https://arxiv.org/abs/2012.11539">Proof Without Words: Sums of Powers of Natural numbers</a>, arXiv:2012.11539 [math.HO], 2020.

%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.

%H Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin, <a href="http://arxiv.org/abs/1505.04479">Peaks Sets of Classical Coxeter Groups</a>, arXiv preprint arXiv:1505.04479 [math.GR], 2015.

%H Michael Dougherty, Christopher French, Benjamin Saderholm, and Wenyang Qian, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/French/french2.html">Hankel Transforms of Linear Combinations of Catalan Numbers</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.1.

%H David Galvin and Courtney Sharpe, <a href="https://arxiv.org/abs/2409.15555">Independent set sequence of linear hyperpaths</a>, arXiv:2409.15555 [math.CO], 2024. See p. 7.

%H Manfred Goebel, <a href="http://dx.doi.org/10.1007/s002000050118">Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials</a>, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.

%H T. Aaron Gulliver, <a href="http://www.m-hikari.com/imf-2011/17-20-2011/index.html">Sequences from hexagonal pyramid of integers</a>, International Mathematical Forum, Vol. 6, 2011, no. 17, p. 821-827.

%H Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

%H Milan Janjic, <a href="https://web.archive.org/web/20190226144349/https://pdfs.semanticscholar.org/801b/6b226bfe1d6b002fb4946c3957d7052132bd.pdf">Two Enumerative Functions</a>.

%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013.

%H R. Jovanovic, <a href="http://web.archive.org/web/20060214203801/http://milan.milanovic.org/math/Math.php?akcija=SviPiram">First 2500 Pyramidal numbers</a>.

%H R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.

%H T. Mansour, <a href="https://arxiv.org/abs/math/0202219">Restricted permutations by patterns of type 2-1</a>, arXiv:math/0202219 [math.CO], 2002.

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

%H Cleve Moler, <a href="http://www.netlib.org/linpack/sgefa.f">LINPACK subroutine sgefa.f</a>, University of New Mexico, Argonne National Lab, 1978.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=DIsW_6u7jrA">Counting on a chessboard.</a>, YouTube video, 2021.

%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7.

%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 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/grid-squares">Square Counting</a>.

%H Think Twice, <a href="https://www.youtube.com/watch?v=aXbT37IlyZQ">Sum of n squares | explained visually |</a>, video (2017).

%H Herman Tulleken, <a href="https://www.researchgate.net/publication/333296614_Polyominoes">Polyominoes 2.2: How they fit together</a>, (2019).

%H G. N. Watson, <a href="http://archive.org/stream/messengerofmathe4849cambuoft#page/n9/mode/2up">The problem of the square pyramid</a>, Messenger of Mathematics 48 (1918), pp. 1-22.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FaulhabersFormula.html">Faulhaber's Formula</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerSum.html">PowerSum</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber&#39;s_formula">Faulhaber's formula</a>.

%H G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on"n^2"</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</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/Tu#2wis">Index entries for two-way infinite sequences</a>.

%F G.f.: x*(1+x)/(1-x)^4.

%F E.g.f.: (x + 3*x^2/2 + x^3/3)*exp(x).

%F a(n) = n*(n+1)*(2*n+1)/6 = binomial(n+2, 3) + binomial(n+1, 3).

%F 2*a(n) = A006331(n). - _N. J. A. Sloane_, Dec 11 1999

%F Can be extended to Z with a(n) = -a(-1-n) for all n in Z.

%F a(n) = binomial(2*(n+1), 3)/4. - _Paul Barry_, Jul 19 2003

%F a(n) = (((n+1)^4 - n^4) - ((n+1)^2 - n^2))/12. - Xavier Acloque, Oct 16 2003

%F From _Alexander Adamchuk_, Oct 26 2004: (Start)

%F a(n) = sqrt(Sum_{j=1..n} Sum_{i=1..n} (i*j)^2).

%F a(n) = (Sum_{k=1..n} Sum_{j=1..n} Sum_{i=1..n} (i*j*k)^2)^(1/3). (End)

%F a(n) = Sum_{i=1..n} i*(2*n-2*i+1); sum of squares gives 1 + (1+3) + (1+3+5) + ... - _Jon Perry_, Dec 08 2004

%F a(n+1) = A000217(n+1) + 2*A000292(n). - _Creighton Dement_, Mar 10 2005

%F Sum_{n>=1} 1/a(n) = 6*(3-4*log(2)); Sum_{n>=1} (-1)^(n+1)*1/a(n) = 6*(Pi-3). - _Philippe Deléham_, May 31 2005

%F Sum of two consecutive tetrahedral (or pyramidal) numbers A000292: C(n+3,3) = (n+1)*(n+2)*(n+3)/6: a(n) = A000292(n-1) + A000292(n). - _Alexander Adamchuk_, May 17 2006

%F Euler transform of length-2 sequence [ 5, -1 ]. - _Michael Somos_, Sep 04 2006

%F a(n) = a(n-1) + n^2. - _Rolf Pleisch_, Jul 22 2007

%F a(n) = A132121(n,0). - _Reinhard Zumkeller_, Aug 12 2007

%F a(n) = binomial(n, 2) + 2*binomial(n, 3). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009, corrected by _M. F. Hasler_, Jan 02 2024

%F a(n) = A168559(n) + 1 for n > 0. - _Reinhard Zumkeller_, Feb 03 2012

%F a(n) = Sum_{i=1..n} J_2(i)*floor(n/i), where J_2 is A007434. - _Enrique Pérez Herrero_, Feb 26 2012

%F a(n) = s(n+1, n)^2 - 2*s(n+1, n-1), where s(n, k) are Stirling numbers of the first kind, A048994. - _Mircea Merca_, Apr 03 2012

%F a(n) = A001477(n) + A000217(n) + A007290(n+2) + 1. - _J. M. Bergot_, May 31 2012

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 2. - _Ant King_, Oct 17 2012

%F a(n) = (A000292(n) + A002411(n))/2. - _Omar E. Pol_, Jan 11 2013

%F a(n) = Sum_{i = 1..n} Sum_{j = 1..n} min(i,j). - _Enrique Pérez Herrero_, Jan 15 2013

%F a(n) = A000217(n) + A007290(n+1). - _Ivan N. Ianakiev_, May 10 2013

%F a(n) = (A047486(n+2)^3 - A047486(n+2))/24. - _Richard R. Forberg_, Dec 25 2013

%F a(n) = Sum_{i=0..n-1} (n-i)*(2*i+1), with a(0) = 0. After 0, row sums of the triangle in A101447. - _Bruno Berselli_, Feb 10 2014

%F a(n) = n + 1 + Sum_{i=1..n+1} (i^2 - 2i). - _Wesley Ivan Hurt_, Feb 25 2014

%F a(n) = A000578(n+1) - A002412(n+1). - _Wesley Ivan Hurt_, Jun 28 2014

%F a(n) = Sum_{i = 1..n} Sum_{j = i..n} max(i,j). - _Enrique Pérez Herrero_, Dec 03 2014

%F a(n) = (2*n^3 + 3*n^2 + n)/6, see Singh (2013). - _Alonso del Arte_, Feb 20 2015

%F For n >= 2, a(n) = A028347(n+1) + A101986(n-2). - _Bui Quang Tuan_, Apr 03 2015

%F For n > 0: a(n) = A258708(n+3,n-1). - _Reinhard Zumkeller_, Jun 23 2015

%F a(n) = A175254(n) + A072481(n), n >= 1. - _Omar E. Pol_, Aug 12 2015

%F a(n) = A000332(n+3) - A000332(n+1). - _Antal Pinter_, Dec 27 2015

%F Dirichlet g.f.: zeta(s-3)/3 + zeta(s-2)/2 + zeta(s-1)/6. - _Ilya Gutkovskiy_, Jun 26 2016

%F a(n) = A080851(2,n-1). - _R. J. Mathar_, Jul 28 2016

%F a(n) = (A005408(n) * A046092(n))/12 = (2*n+1)*(2*n*(n+1))/12. - _Bruce J. Nicholson_, May 18 2017

%F 12*a(n) = (n+1)*A001105(n) + n*A001105(n+1). - _Bruno Berselli_, Jul 03 2017

%F a(n) = binomial(n-1, 1) + binomial(n-1, 2) + binomial(n, 3) + binomial(n+1, 2) + binomial(n+1, 3). - _Tony Foster III_, Aug 24 2018

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Nathan Fox_, Dec 04 2019

%F Let T(n) = A000217(n), the n-th triangular number.  Then a(n) = (T(n)+1)^2 + (T(n)+2)^2 + ... + (T(n)+n)^2 - (n+2)*T(n)^2. - _Charlie Marion_, Dec 31 2019

%F a(n) = 2*n - 1 - a(n-2) + 2*a(n-1). - _Boštjan Gec_, Nov 09 2023

%e G.f. = x + 5*x^2 + 14*x^3 + 30*x^4 + 55*x^5 + 91*x^6 + 140*x^7 + 204*x^8 + ...

%p A000330 := n -> n*(n+1)*(2*n+1)/6;

%p a := n->(1/6)*n*(n+1)*(2*n+1): seq(a(n),n=0..53); # _Emeric Deutsch_

%p A000330 := (1+z)/(z-1)^4; # _Simon Plouffe_ (in his 1992 dissertation: generating function for sequence starting at a(1))

%p with(combstruct): ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m*2), m=1..45) ; # _Zerinvary Lajos_, Jan 02 2008

%p a := n -> sum(k^2, k=1..n):seq(a(n), n=0...44); # _Zerinvary Lajos_, Jun 15 2008

%p nmax := 44; for n from 0 to nmax do fz(n) := product( (1-(2*m-1)*z)^(n+1-m) , m=1..n); c(n) := abs(coeff(fz(n),z,1)); end do: a := n-> c(n): seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Mar 07 2009

%t Table[Binomial[w+2, 3] + Binomial[w+1, 3], {w, 0, 30}]

%t CoefficientList[Series[x(1+x)/(1-x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 30 2014 *)

%t Accumulate[Range[0,50]^2] (* _Harvey P. Dale_, Sep 25 2014 *)

%o (PARI) {a(n) = n * (n+1) * (2*n+1) / 6};

%o (PARI) upto(n) = [x*(x+1)*(2*x+1)/6 | x<-[0..n]] \\ _Cino Hilliard_, Jun 18 2007, edited by _M. F. Hasler_, Jan 02 2024

%o (Haskell)

%o a000330 n = n * (n + 1) * (2 * n + 1) `div` 6

%o a000330_list = scanl1 (+) a000290_list

%o -- _Reinhard Zumkeller_, Nov 11 2012, Feb 03 2012

%o (Maxima) A000330(n):=binomial(n+2,3)+binomial(n+1,3)$

%o makelist(A000330(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */

%o (Magma) [n*(n+1)*(2*n+1)/6: n in [0..50]]; // _Wesley Ivan Hurt_, Jun 28 2014

%o (Magma) [0] cat [((2*n+3)*Binomial(n+2,2))/3: n in [0..40]]; // _Vincenzo Librandi_, Jul 30 2014

%o (Python) a=lambda n: (n*(n+1)*(2*n+1))//6 # _Indranil Ghosh_, Jan 04 2017

%o (Sage) [n*(n+1)*(2*n+1)/6 for n in (0..30)] # _G. C. Greubel_, Dec 31 2019

%o (GAP) List([0..30], n-> n*(n+1)*(2*n+1)/6); # _G. C. Greubel_, Dec 31 2019

%Y Cf. A000217, A000292, A000537, A005408, A006003, A006331, A033994, A033999, A046092, A050409, A050446, A050447, A100157, A132124, A132112, A156921, A157702, A258708, A351830.

%Y Sums of 2 consecutive terms give A005900.

%Y Column 0 of triangle A094414.

%Y Column 1 of triangle A008955.

%Y Right side of triangle A082652.

%Y Row 2 of array A103438.

%Y Partial sums of A000290.

%Y Cf. similar sequences listed in A237616 and A254142.

%Y Cf. |A084930(n, 1)|.

%Y Cf. A253903 (characteristic function).

%Y Cf. A034705 (differences of any two terms).

%K nonn,easy,core,nice

%O 0,3

%A _N. J. A. Sloane_

%E Partially edited by _Joerg Arndt_, Mar 11 2010