|
%I M3844 N1574
%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: 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
%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 You (giawgwan(AT)single.url.com.tw), 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, R.H.S. of relation (i) telescopes to f(n+1)+f(n) = T(n)*{(n+2)+(n-1)}, whence result sum_(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. 7, 2004.
%C Number of triples of integers from {1,2,...,n} whose last component is greater than or equal to the others.
%C Kekule numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
%C Euler transform of length 2 sequence [ 5, -1]. - _Michael Somos_, Sep 04 2006
%C Sum of the first n squares, or square pyramidal numbers. - Cino Hilliard (hillcino368(AT)hotmail.com), 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 (sfalcon(AT)dma.ulpgc.es), Feb 05 2008
%C Hankel transform of C(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,C(n+2,i+2)*b(i)}, where b(i)=1,2,0,0,0,... [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%C Convolution of A001477 with A005408: a(n)=SUM((2*k+1)*(n-k):0<=k<=n). [From _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(A000217(i), i=0..n-1) and this is the case d=1 in the identity n^2*(d*n-d+2)/2-sum(i*(d*i-d+2)/2, i=0..n-1) = 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*(i+2*d+1)/2, i=0..n-1) = 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. number k = 195 is quadratic mean (root mean square) of first 337 positive integers. There are other such numbers - see A084231 and A084232. [From Jaroslav Krizek, May 23 2010]
%C Contribution from Carmine Suriano, Sep 10 2010: (Start)
%C Also the number of moves to solve the "alternate coins game".
%C 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. (End)
%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 for n is 60 times the numbers in this sequence. Using three consecutive odd numbers a, b, c, (a+b+c)^3 - (a^3 + b^3 + c^3) equals 576=24^2 times the numbers in this sequence. [J. M. Bergot, Aug 23 2011]
%C Contribution from _Ant King_, Oct 17 2012: (Start)
%C For n>0, the digital roots of this sequence A010888(A000330(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(A000330(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}
%C (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
%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 B. Babcock and A. van Tuyl, Revisiting the spreading and covering numbers, Arxiv preprint arXiv:1109.5847, 2011
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, 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, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
%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, pg 293.
%D Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%D M. Merca, A Special Case of the Generalized Girard-Waring Formula, Journal of Integer Sequences, Vol. 15 (2012), #12.5.7. - From _N. J. A. Sloane_, Nov 25 2012
%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).
%H T. D. Noe, <a href="/A000330/b000330.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H B. 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 H. Bottomley, <a href="/A000330/a000330.gif">Illustration of initial terms</a>
%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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviPiram">First 2500 Pyramidal numbers</a> [Broken link?]
%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted permutations by patterns of type 2-1</a>.
%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 _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/grid-squares">Square Counting</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FaulhabersFormula.html">Faulhaber's Formula</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber's_formula">Faulhaber's formula</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</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/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(1+x)/(1-x)^4.
%F E.g.f.: (x+3/2*x^2+1/3*x^3)*exp(x).
%F a(n) = n*(n+1)*(2*n+1)/6 = binomial(n+2, 3)+binomial(n+1, 3)
%F a(n) = -a(-1-n).
%F 2*a(n) = A006331(n). - _N. J. A. Sloane_, Dec 11 1999
%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 a(n) = sqrt(sum(sum[(i*j)^2, {i, 1, n}), {j, 1, n})). a(n) = sum(sum(sum((i*j*k)^2, {i, 1, n}), {j, 1, n}), {k, 1, n})^(1/3). - _Alexander Adamchuk_, Oct 26 2004
%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-1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 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-2). - _Alexander Adamchuk_, May 17 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 Starting n (-1,0,1,2,...), a(n) = C(n+2,2)+2*C(n+2,3). [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%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. [From _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
%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, 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 (zerinvarylajos(AT)yahoo.com), Jan 02 2008
%p a:=n->sum(k^2, k=1..n):seq(a(n), n=0...44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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, 1, 30}]
%o (PARI) a(n)=n*(n+1)*(2*n+1)/6
%o (PARI) sumsq(n) = for(x=0,n,y=x*(x+1)*(2*x+1)/6;(print1(y","))) - Cino Hilliard (hillcino368(AT)hotmail.com), Jun 18 2007
%o (PARI) a(n)=sum(m=1,n,sum(i=1,m,(2*i-1))) - _Alexander R. Povolotsky_, Nov 04 2007
%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 */
%Y Cf. A000217, A050446, A050447, A000537, A006003, A006331.
%Y Cf. A000292, A033994, A132124, A132112, A050409.
%Y Sums of 2 consecutive terms give A005900.
%Y Column 0 of triangle A094414. Column 1 of triangle A008955. Right side of triangle A082652. Row 2 of array A103438.
%Y Partial sums of A000290. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%Y Cf. A156921, A157702.
%K nonn,easy,core,nice,changed
%O 0,3
%A _N. J. A. Sloane_.
%E Partially edited by _Joerg Arndt_, Mar 11 2010
|
