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A000330 Square pyramidal numbers: 0^2 + 1^2 + 2^2 +...+ n^2 = n*(n+1)*(2*n+1)/6.
(Formerly M3844 N1574)
291
0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, 16206, 17575, 19019, 20540, 22140, 23821, 25585, 27434, 29370 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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 of Watson. - Charles R Greathouse IV, Jun 21 2013]

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

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

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

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

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

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 result sum_(1, n)r^2 = n*(n+1)*(2*n+1)/6 immediately follows. - Lekraj Beedassy, Aug 06 2004

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.

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

Kekule numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

Euler transform of length 2 sequence [ 5, -1]. - Michael Somos, Sep 04 2006

Sum of the first n squares, or square pyramidal numbers. - Cino Hilliard, Jun 18 2007

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

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

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

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

Convolution of A000290 with A000012. - Sergio Falcon, Feb 05 2008

Hankel transform of C(2*n-3,n-1) is -a(n). - Paul Barry, Feb 12 2008

Starting (1, 5, 14, 30,...) = binomial transform of [1, 4, 5, 2, 0, 0, 0,...]. - Gary W. Adamson, Jun 13 2008

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,... - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

Convolution of A001477 with A005408: a(n)=SUM((2*k+1)*(n-k):0<=k<=n). - Reinhard Zumkeller, Mar 07 2009

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

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

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. - Jaroslav Krizek, May 23 2010

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

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

From Ant King, Oct 17 2012: (Start)

. 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}

. 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)

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

Sum of entries of n X n square matrix with elements min(i,j). - Enrique Pérez Herrero, Jan 16 2013

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

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

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

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 displayed 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

Beukers and Top prove that no square pyramidal number > 1 equals a tetrahedral number A000292. - Jonathan Sondow, Jun 21 2014

REFERENCES

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.

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

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

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

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.

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

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

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.

M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pg 293.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

B. Babcock and A. van Tuyl, Revisiting the spreading and covering numbers, Arxiv preprint arXiv:1109.5847, 2011

B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).

F. Beukers and J. Top, On oranges and integral points on certain plane cubic curves, Nieuw Arch. Wiskd., IV (1988), Ser. 6, No. 3, 203-210.

H. Bottomley, Illustration of initial terms

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.

T. Aaron Gulliver, Sequences from hexagonal pyramid of integers, International Mathematical Forum, Vol. 6, 2011, no. 17, p. 821 - 827.

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013.

R. Jovanovic, First 2500 Pyramidal numbers [Broken link?]

T. Mansour, Restricted permutations by patterns of type 2-1.

Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

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.

T. Sillke, Square Counting

G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.

Eric Weisstein's World of Mathematics, Faulhaber's Formula

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Wikipedia, Faulhaber's formula

G. Xiao, Sigma Server, Operate on"n^2"

Index entries for "core" sequences

Index entries for two-way infinite sequences

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

FORMULA

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

E.g.f.: (x+3/2*x^2+1/3*x^3)*exp(x).

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

a(n) = -a(-1-n).

2*a(n) = A006331(n). - N. J. A. Sloane, Dec 11 1999

a(n) = binomial(2*(n+1), 3)/4. - Paul Barry, Jul 19 2003

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

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

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

a(n+1) = A000217(n+1) + 2*A000292(n). - Creighton Dement, Mar 10 2005

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

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

a(n) = a(n-1) + n^2. - Rolf Pleisch, Jul 22 2007

a(n) = A132121(n,0). - Reinhard Zumkeller, Aug 12 2007

Starting n (-1,0,1,2,...), a(n) = C(n+2,2)+2*C(n+2,3). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

a(n) = A168559(n) + 1 for n > 0. - Reinhard Zumkeller, Feb 03 2012

a(n) = sum(i=1..n, J_2(i)*floor(n/i)), where J_2 is A007434. - Enrique Pérez Herrero, Feb 26 2012

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

a(n) = A001477(n) + A000217(n) + A007290(n+2) + 1. - J. M. Bergot, May 31 2012

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2. - Ant King Oct 17 2012

a(n) = (A000292(n) + A002411(n))/2. - Omar E. Pol, Jan 11 2013

a(n) = sum(i=1..n, sum(j=1..n, min(i,j))). - Enrique Pérez Herrero, Jan 15 2013

a(n) = A000217(n) + A007290(n+1). - Ivan N. Ianakiev, May 10 2013

a(n) = (A047486(n+2)^3 - A047486(n+2))/24. - Richard R. Forberg, Dec 25 2013

a(n) = sum( (n-i)*(2*i+1), i=0..n-1 ), with a(0)=0. After 0, row sums of the triangle in A101447. [Bruno Berselli, Feb 10 2014]

a(n) = n + 1 + sum_{i=1..n+1} (i^2 - 2i). - Wesley Ivan Hurt, Feb 25 2014

a(n) = A000578(n+1) - A002412(n+1). - Wesley Ivan Hurt, Jun 28 2014

EXAMPLE

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

MAPLE

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

a:=n->(1/6)*n*(n+1)*(2*n+1): seq(a(n), n=0..53); # Emeric Deutsch

A000330:=(1+z)/(z-1)^4; # Simon Plouffe (in his 1992 dissertation, sequence starting at a(1))

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

a:=n->sum(k^2, k=1..n):seq(a(n), n=0...44); # Zerinvary Lajos, Jun 15 2008

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

MATHEMATICA

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

CoefficientList[Series[x (1 + x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

PROG

(PARI) a(n)=n*(n+1)*(2*n+1)/6;

(PARI) sumsq(n) = for(x=0, n, y=x*(x+1)*(2*x+1)/6; (print1(y", "))) \\ Cino Hilliard, Jun 18 2007

(PARI) a(n)=sum(m=1, n, sum(i=1, m, (2*i-1))) \\ Alexander R. Povolotsky, Nov 04 2007

(Haskell)

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

a000330_list = scanl1 (+) a000290_list

-- Reinhard Zumkeller, Nov 11 2012, Feb 03 2012

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

makelist(A000330(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */

(MAGMA) [n*(n+1)*(2*n+1)/6: n in [0..50]]; // Wesley Ivan Hurt, Jun 28 2014

(MAGMA) [0] cat [((2*n+3)*Binomial(n+2, 2))/3: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

CROSSREFS

Cf. A000217, A050446, A050447, A000537, A006003, A006331, A000292, A033994, A132124, A132112, A050409, A156921, A157702.

Sums of 2 consecutive terms give A005900.

Column 0 of triangle A094414. Column 1 of triangle A008955. Right side of triangle A082652. Row 2 of array A103438.

Partial sums of A000290.

Cf. similar sequences listed in A237616.

Cf. |A084930(n, 1)| - Wolfdieter Lang, Aug 05 2014

Sequence in context: A231685 A074784 A109678 * A211804 A238604 A166068

Adjacent sequences:  A000327 A000328 A000329 * A000331 A000332 A000333

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Partially edited by Joerg Arndt, Mar 11 2010

STATUS

approved

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Last modified August 29 18:25 EDT 2014. Contains 246200 sequences.