A008911 -id:A008911 - cp's OEIS frontend

A002415 4-dimensional pyramidal numbers: a(n) = n^2*(n^2-1)/12.

0, 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, 111265, 124950, 139860, 156066, 173641, 192660, 213200, 235340

Offset: 0

N. J. A. Sloane

Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2) for 2 clumps of 2 parentheses and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217.
E.g., for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).
Let M_n denote the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre, Nov 09 2002
Let M_n denote the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002 [See A114327 for the infinite matrix M in triangular form. - Wolfdieter Lang, Feb 05 2018]
Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki, Aug 26 2004
Number of tilings of a <2,n,2> hexagon.
a(n) is the number of squares of side length at least 1 having vertices at the points of an n X n unit grid of points (the vertices of an n-1 X n-1 chessboard). [For a proof, see Comments in A051602. - N. J. A. Sloane, Sep 29 2021] For example, on the 3 X 3 grid (the vertices of a 2 X 2 chessboard) there are four 1 X 1 squares, one (skew) sqrt(2) X sqrt(2) square, and one 3 X 3 square, so a(3)=6. On the 4 X 4 grid (the vertices of a 3 X 3 chessboard) there are 9 1 X 1 squares, 4 2 X 2 squares, 1 3 X 3 square, 4 sqrt(2) X sqrt(2) squares, and 2 sqrt(5) X sqrt(5) squares, so a(4) = 20. See also A024206, A108279. [Comment revised by N. J. A. Sloane, Feb 11 2015]
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith, Apr 24 2006
a(n) is the number of 4 X 4 matrices (symmetrical about each diagonal) M = [a,b,c,d;b,e,f,c;c,f,e,b;d,c,b,a] with a+b+c+d=b+e+f+c=n+2; (a,b,c,d,e,f natural numbers). - Philippe Deléham, Apr 11 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 5-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
a(n) is the number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan, Sep 20 2007
Starting (1,6,20,50,...) = third partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} C(n+3,i+3)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
4-dimensional square numbers. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
Equals row sums of triangle A177877; a(n), n > 1 = (n-1) terms in (1,2,3,...) dot (...,3,2,1) with additive carryovers. Example: a(4) = 20 = (1,2,3) dot (3,2,1) with carryovers = (1*3) + (2*2 + 3) + (3*1 + 7) = (3 + 7 + 10).
Convolution of the triangular numbers A000217 with the odd numbers A004273.
a(n+2) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and w-x=max{w,x,y,z}-min{w,x,y,z}. - Clark Kimberling, May 28 2012
The second level of finite differences is a(n+2) - 2*a(n+1) + a(n) = (n+1)^2, the squares. - J. M. Bergot, May 29 2012
Because the differences of this sequence give A000330, this is also the number of squares in an n+1 X n+1 grid whose sides are not parallel to the axes.
a(n+2) gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing. - Jon Perry, Mar 30 2013
For n consecutive numbers 1,2,3,...,n, the sum of all ways of adding the k-tuples of consecutive numbers for n=a(n+1). As an example, let n=4: (1)+(2)+(3)+(4)=10; (1+2)+(2+3)+(3+4)=15; (1+2+3)+(2+3+4)=15; (1+2+3+4)=10 and the sum of these is 50=a(4+1)=a(5). - J. M. Bergot, Apr 19 2013
If P(n,k) = n*(n+1)*(k*n-k+3)/6 is the n-th (k+2)-gonal pyramidal number, then a(n) = P(n,k)*P(n-1,k-1) - P(n-1,k)*P(n,k-1). - Bruno Berselli, Feb 18 2014
For n > 1, a(n) = 1/6 of the area of the trapezoid created by the points (n,n+1), (n+1,n), (1,n^2+n), (n^2+n,1). - J. M. Bergot, May 14 2014
For n > 3, a(n) is twice the area of a triangle with vertices at points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), and (C(n+2,4),C(n+3,4)). - J. M. Bergot, Jun 03 2014
a(n) is the dimension of the space of metric curvature tensors (those having the symmetries of the Riemann curvature tensor of a metric) on an n-dimensional real vector space. - Daniel J. F. Fox, Dec 15 2018
Coefficients in the terminating series identity 1 - 6*n/(n + 5) + 20*n*(n - 1)/((n + 5)*(n + 6)) - 50*n*(n - 1)*(n - 2)/((n + 5)*(n + 6)*(n + 7)) + ... = 0 for n = 1,2,3,.... Cf. A000330 and A005585. - Peter Bala, Feb 18 2019

			a(7) = 6*21 - (6*0 + 4*1 + 2*3 + 0*6 - 2*10 - 4*15) = 196. - _Bruno Berselli_, Jun 22 2013
G.f. = x^2 + 6*x^3 + 20*x^4 + 50*x^5 + 105*x^6 + 196*x^7 + 336*x^8 + ...

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
Henry Bottomley, Illustration of initial terms
Duane DeTemple, Using Squares to Sum Squares, The College Mathematics Journal, ? (2010), 214-221.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Jones, S. Kitaev, and J. Remmel, Frame patterns in n-cycles, arXiv preprint arXiv:1311.3332 [math.CO], 2013.
Sandi Klavžar, Balázs Patkós, Gregor Rus, and Ismael G. Yero, On general position sets in Cartesian grids, arXiv:1907.04535 [math.CO], 2019.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
Calvin Lin, Squares on a grid, April 2015
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 9, 13, 15, 25.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698).
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138. See Table I.
Royce A. Speck, The Number of Squares on a Geoboard, School Science and Mathematics, Volume 79, Issue 2, pages 145-150, February 1979
Eric Weisstein's World of Mathematics, Square Tiling.
Eric Weisstein's World of Mathematics, Riemann Tensor.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n) = ((-1)^n)*A053120(2*n, 4)/8 (one-eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.
Second row of array A103905.
Third column of Narayana numbers A001263.
Cf. A001079, A006011, A008911, A037270, A047819, A047928, A071253, A083374, A107891, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A108279, A024206.
Partial sums of A000330.
The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers (A000027) with the k-gonal numbers.
Cf. A002388, A051602, A114327.

List([0..45],n->Binomial(n^2,2)/6); # Muniru A Asiru, Dec 15 2018

[n^2*(n^2-1)/12: n in [0..50]]; // Wesley Ivan Hurt, May 14 2014

A002415 := proc(n) binomial(n^2,2)/6 ; end proc: # Zerinvary Lajos, Jan 07 2008

Table[(n^4 - n^2)/12, {n, 0, 40}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,1,6,20},40] (* Harvey P. Dale, Nov 29 2011 *)

PARI
```
a(n) = n^2 * (n^2 - 1) / 12;
```

x='x+O('x^200); concat([0, 0], Vec(x^2*(1+x)/(1-x)^5)) \\ Altug Alkan, Mar 23 2016

G.f.: x^2*(1+x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{i=0..n} (n-i)*i^2 = a(n-1) + A000330(n-1) = A000217(n)*A000292(n-2)/n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3, convolution of the natural numbers with the squares. - Henry Bottomley, Oct 19 2000
a(n)+1 = A079034(n). - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003
a(n) = 2*C(n+2, 4) - C(n+1, 3). - Paul Barry, Mar 04 2003
a(n) = C(n+2, 4) + C(n+1, 4). - Paul Barry, Mar 13 2003
a(n) = Sum_{k=1..n} A000330(n-1). - Benoit Cloitre, Jun 15 2003
a(n) = n*C(n+1,3)/2 = C(n+1,3)*C(n+1,2)/(n+1). - Mitch Harris, Jul 06 2006
a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n)/6. - Zerinvary Lajos, May 09 2007
a(n) = (1/2)*Sum_{1 <= x_1, x_2 <= n} (det V(x_1,x_2))^2 = (1/2)*Sum_{1 <= i,j <= n} (i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = C(n+1,3) + 2*C(n+1,4). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (1/48)*sinh(2*arccosh(n))^2. - Artur Jasinski, Feb 10 2010
a(n) = n*A000292(n-1)/2. - Tom Copeland, Sep 13 2011
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 29 2011
a(n) = (n-1)*A000217(n-1) - Sum_{i=0..n-2} (n-1-2*i)*A000217(i) for n > 1. - Bruno Berselli, Jun 22 2013
a(n) = C(n,2)*C(n+1,3) - C(n,3)*C(n+1,2). - J. M. Bergot, Sep 17 2013
a(n) = Sum_{k=1..n} ( (2k-n)* k(k+1)/2 ). - Wesley Ivan Hurt, Sep 26 2013
a(n) = floor(n^2/3) + 3*Sum_{k=1..n} k^2*floor((n-k+1)/3). - Mircea Merca, Feb 06 2014
Euler transform of length 2 sequence [6, -1]. - Michael Somos, May 28 2014
G.f. x^2*2F1(3,4;2;x). - R. J. Mathar, Aug 09 2015
Sum_{n>=2} 1/a(n) = 21 - 2*Pi^2 = 1.260791197821282762331... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A080852(2,n-2). - R. J. Mathar, Jul 28 2016
a(n) = A046092(n) * A046092(n-1)/48 = A000217(n) * A000217(n-1)/3. - Bruce J. Nicholson, Jun 06 2017
E.g.f.: (1/12)*exp(x)*x^2*(6 + 6*x + x^2). - Stefano Spezia, Dec 07 2018
Sum_{n>=2} (-1)^n/a(n) = Pi^2 - 9 (See A002388). - Amiram Eldar, Jun 28 2020

Typo in link fixed by Matthew Vandermast, Nov 22 2010

Redundant comment deleted and more detail on relationship with A000330 added by Joshua Zucker, Jan 01 2013

A083374 a(n) = n^2 * (n^2 - 1)/2.

Original entry on oeis.org

0, 6, 36, 120, 300, 630, 1176, 2016, 3240, 4950, 7260, 10296, 14196, 19110, 25200, 32640, 41616, 52326, 64980, 79800, 97020, 116886, 139656, 165600, 195000, 228150, 265356, 306936, 353220, 404550, 461280, 523776, 592416, 667590, 749700, 839160, 936396

Offset: 1

Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003

Triangular numbers t_n as n runs through the squares.
Partial sums of A055112: If one generated Pythagorean primitive triangles from n, n+1, then the collective areas of n of them would be equal to the numbers in this sequence. The sum of the first three triangles is 6+30+84 = 120 which is the third nonzero term of the sequence. - J. M. Bergot, Jul 14 2011
Second leg of Pythagorean triangles with smallest side a cube: A000578(n)^2 + a(n)^2 = A037270(n)^2. - Martin Renner, Nov 12 2011
a(n) is the number of segments on an n X n grid or geoboard. - Martin Renner, Apr 17 2014
Consider the partitions of 2n into two parts (p,q). Then a(n) is the total volume of the family of rectangular prisms with dimensions p, q and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000578, A002415, A006002, A006011, A008911, A037270, A047928, A055112.

[n^2*(n^2-1)/2: n in [1..40]]; // Vincenzo Librandi, Sep 14 2011

A000217:=func; [&+[k*A000217(2*k+1): k in [0..n-1]]: n in [1..40]]; // Bruno Berselli, Sep 04 2013

Maple

A083374 := proc(n) n^2*(n^2-1)/2 ; end proc: # R. J. Mathar, Aug 23 2011

Mathematica

Table[n^2*(n^2-1)/2, {n,40}] (* T. D. Noe, Oct 25 2006 *)

PARI

a(n)=binomial(n^2,2) \\ Charles R Greathouse IV, Aug 23 2011

Formula

a(n) = (n + 1) * A006002(n).

a(n) = A047928(n)/2 = A002415(n+1)*6 = A006011(n+1)*2 = A008911(n+1)*3. - Zerinvary Lajos, May 09 2007

a(n) = binomial(n^2,2), n>=1. - Zerinvary Lajos, Jan 07 2008

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5. - R. J. Mathar, Apr 10 2009

G.f.: -6*x^2*(1+x)/(x-1)^5. - R. J. Mathar, Apr 10 2009

Sum_{n>1} 1/a(n) = (21 - 2*Pi^2)/6. - Enrique Pérez Herrero, Apr 01 2013

a(n) = Sum_{k=0..n-1} k*A000217(2*k+1). - Bruno Berselli, Sep 04 2013

a(n) = 2*A000217(n-1)*A000217(n). - Gionata Neri, May 04 2015

a(n) = Sum_{i=1..n^2-1} i. - Wesley Ivan Hurt, Nov 24 2015

E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/2. - Stefano Spezia, Jun 06 2021

Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 3/2. - Amiram Eldar, Nov 02 2021

Extensions

Corrected and extended by T. D. Noe, Oct 25 2006

A006011 a(n) = n^2*(n^2 - 1)/4.

Original entry on oeis.org

0, 0, 3, 18, 60, 150, 315, 588, 1008, 1620, 2475, 3630, 5148, 7098, 9555, 12600, 16320, 20808, 26163, 32490, 39900, 48510, 58443, 69828, 82800, 97500, 114075, 132678, 153468, 176610, 202275, 230640, 261888, 296208, 333795, 374850, 419580, 468198

Offset: 0

N. J. A. Sloane, Simon Plouffe

Products of two consecutive triangular numbers (A000217).
a(n) is the number of Lyndon words of length 4 on an n-letter alphabet. A Lyndon word is a primitive word that is lexicographically earliest in its cyclic rotation class. For example, a(2)=3 counts 1112, 1122, 1222. - David Callan, Nov 29 2007
For n >= 2 this is the second rightmost column of A163932. - Johannes W. Meijer, Oct 16 2009
Partial sums of A059270. - J. M. Bergot, Jun 27 2013
Using the integers, triangular numbers, and squares plot the points (A001477(n),A001477(n+1)), (A000217(n), A000217(n+1)), and (A000290(n),A000290(n+1)) to create the vertices of a triangle. One-half the area of this triangle = a(n). - J. M. Bergot, Aug 01 2013
a(n) is the Wiener index of the triangular graph T(n+1). - Emeric Deutsch, Aug 26 2013

			From _Bruno Berselli_, Aug 29 2014: (Start)
After the zeros, the sequence is provided by the row sums of the triangle:
   3;
   4, 14;
   5, 16, 39;
   6, 18, 42,  84;
   7, 20, 45,  88, 155;
   8, 22, 48,  92, 160, 258;
   9, 24, 51,  96, 165, 264, 399;
  10, 26, 54, 100, 170, 270, 406, 584;
  11, 28, 57, 104, 175, 276, 413, 592, 819;
  12, 30, 60, 108, 180, 282, 420, 600, 828, 1110; etc.,
where T(r,c) = c*(c^2+r+1), with r = row index, c = column index, r >= c > 0. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Miguel Azaola and Francisco Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., Vol. 27 (2002), pp. 29-48 (see Prop. 4.2(a)).
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Triangular Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Thrice A002415. Row 4 of A074650.
Cf. A000217, A000290, A000537, A001477, A002415, A008911, A047928, A059270, A083374, A126274, A163932, A228317
A column of A124428.

[n^2*(n^2-1)/4: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

A006011 := proc(n)
    n^2*(n^2-1)/4 ;
end proc: # R. J. Mathar, Nov 29 2015

Table[n^2 (n^2 - 1)/4, {n, 0, 38}]
Binomial[Range[20]^2, 2]/2 (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 18, 60, 150}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[-3 x (1 + x)/(-1 + x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)
Join[{0},Times@@@Partition[Accumulate[Range[0,40]],2,1]] (* Harvey P. Dale, Aug 08 2025 *)

a(n)=binomial(n^2,2)/2 \\ Charles R Greathouse IV, Jun 27 2013

G.f.: 3*(1 + x) / (1 - x)^5.
a(n) = (n-1)*n/2 * n*(n+1)/2 = A000217(n-1)*A000217(n) = 1/2*(n^2-1)*n^2/2 = 1/2*A000217(n^2-1). - Alexander Adamchuk, Apr 13 2006
a(n) = 3*A002415(n) = A047928(n-1)/4 = A083374(n-1)/2 = A008911(n)*3/2. - Zerinvary Lajos, May 09 2007
a(n) = (A126274(n) - A000537(n+1))/2. - Enrique Pérez Herrero, Mar 11 2013
Ceiling(sqrt(a(n)) + sqrt(a(n-1)))/2 = A000217(n). - Richard R. Forberg, Aug 14 2013
a(n) = Sum_{i=1..n-1} i*(i^2+n) for n > 1 (see Example section). - Bruno Berselli, Aug 29 2014
Sum_{n>=2} 1/a(n) = 7 - 2*Pi^2/3 = 0.42026373260709425411... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n^2+n) - A000217(n)*A000217(n+1). - Charlie Marion, Feb 15 2020
Sum_{n>=2} (-1)^n/a(n) = Pi^2/3 - 3. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/4. - Stefano Spezia, Mar 12 2024

A047659 Number of ways to place 3 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 24, 204, 1024, 3628, 10320, 25096, 54400, 107880, 199400, 348020, 579264, 926324, 1431584, 2148048, 3141120, 4490256, 6291000, 8656860, 11721600, 15641340, 20597104, 26797144, 34479744, 43915768, 55411720, 69312516, 86004800, 105919940

Offset: 0

Paul Zimmermann

Lucas mentions that the number of ways of placing p <= n non-attacking queens on an n X n chessboard is given by a polynomial in n of degree 2p and attribute the result to Mantel, professor in Delft. Cf. Stanley, exercise 15.

E. Landau, Naturwissenschaftliche Wochenschrift (Aug. 2 1896).
R. P. Stanley, Enumerative Combinatorics, vol. I, exercise 15 in chapter 4 (and its solution) asks one to show the existence of a rational generating function for the number of ways of placing k non-attacking queens on an n X n chessboard.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, Jan 26 2013. - _N. J. A. Sloane_, Feb 16 2013
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv:1609.00853 [math.CO], Sep 03 2016.
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 11.
Edmund Landau, Ueber das Achtdamenproblem und seine Verallgemeinerung, Naturwissenschaftliche Wochenschrift, Aug 02 1896.
Edouard Lucas, Récréations mathématiques, Gauthier-Villars, Paris, 1882-1894, Vol. I, p. 228.
Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, P(3) at p.8-9.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Wenxi Wang, Muhammad Usman, Alyas Almaawi, Kaiyuan Wang, Kuldeep S. Meel, Sarfraz Khurshid, A Study of Symmetry Breaking Predicates and Model Counting, National University of Singapore (2020).
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A036464, A061994, A108792, A176186, A178721.

Column k=3 of A348129.

[(3*(2*n-1)*(-1)^n +4*n^6 -40*n^5 +158*n^4 -300*n^3 +264*n^2 -86*n +3)/24: n in [0..35]]; // Vincenzo Librandi, Sep 21 2015

f:=n-> n^6/6 - 5*n^5/3 + 79*n^4/12 - 25*n^3/2 + 11*n^2 - 43*n/12 + 1/8 + (-1)^n*(n/4 - 1/8); [seq(f(n),n=1..40)]; # N. J. A. Sloane, Feb 16 2013

Table[If[EvenQ[n],n (n-2)^2 (2n^3-12n^2+23n-10)/12,(n-1)(n-3) (2n^4- 12n^3+25n^2-14n+1)/12],{n,0,30}] (* or *) LinearRecurrence[ {5,-8,0,14,-14,0,8,-5,1},{0,0,0,0,24,204,1024,3628,10320},30] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=if(n%2, (n - 1)*(n - 3)*(2*n^4 - 12*n^3 + 25*n^2 - 14*n + 1), n*(n - 2)^2*(2*n^3 - 12*n^2 + 23*n - 10))/12 \\ Charles R Greathouse IV, Feb 09 2017

a(n) = n(n - 2)^2(2n^3 - 12n^2 + 23n - 10)/12 if n is even and (n - 1)(n - 3)(2n^4 - 12n^3 + 25n^2 - 14n + 1)/12 if n is odd (Landau, 1896).
a(n) = 5a(n - 1) - 8a(n - 2) + 14a(n - 4) - 14a(n - 5) + 8a(n - 7) - 5a(n - 8) + a(n - 9) for n >= 9.
G.f.: 4(9*x^4 + 35*x^3 + 49*x^2 + 21*x + 6)*x^4/((1 - x)^7*(1 + x)^2).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24, a(5)=204, a(6)=1024, a(7)=3628, a(8)=10320, a(n) = 5*a(n-1)-8*a(n-2)+14*a(n-4)-14*a(n-5)+8*a(n-7)- 5*a(n-8)+ a(n-9). - Harvey P. Dale, Nov 06 2011
a(n) = n^6/6 - 5*n^5/3 + 79*n^4/12 - 25*n^3/2 + 11*n^2 - 43*n/12 + 1/8 + (-1)^n*(n/4 - 1/8) [Chaiken et al.]. - N. J. A. Sloane, Feb 16 2013
a(n) = (3*(2*n-1)*(-1)^n +4*n^6 -40*n^5 +158*n^4 -300*n^3 +264*n^2 -86*n +3)/24. - Antal Pinter, Oct 03 2014
E.g.f.: (exp(2*x)*(3 - 6*x^2 + 8*x^3 + 18*x^4 + 20*x^5 + 4*x^6) -3 - 6*x) / (24*exp(x)). - Vaclav Kotesovec, Feb 15 2015
For n>3, a(n) = A179058(n) -4*(n-2)*A000914(n-2) -2*(n-2)*A002415(n-1) + 2*A008911(n-1) +8*(A001752(n-4) +A007009(n-3)). - Antal Pinter, Sep 20 2015
In general, for m <= n, n >= 3, the number of ways to place 3 nonattacking queens on an m X n board is n^3/6*(m^3 - 3*m^2 + 2*m) - n^2/2*(3*m^3 - 9*m^2 + 6*m) + n/6*(2*m^4 + 20*m^3 - 77*m^2 + 58*m) - 1/24*(39*m^4 - 82*m^3 - 36*m^2 + 88*m) + 1/16*(2*m - 4*n + 1)*(1 + (-1)^(m+1)) + 1/2*(1 + abs(n - 2*m + 3) - abs(n - 2*m + 4))*(1/24*((n - 2*m + 11)^4 - 42*(n - 2*m + 11)^3 + 656*(n - 2*m + 11)^2 - 4518*(n - 2*m + 11) + 11583) - 1/16*(4*m - 2*n - 1)*(1 + (-1)^(n+1))) [Panos Louridas, idee & form 93/2007, pp. 2936-2938]. - Vaclav Kotesovec, Feb 20 2016

The formula given in the Rivin et al. paper is wrong.

Entry improved by comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 30 2001

A047928 a(n) = n(n-1)^2(n-2).

Original entry on oeis.org

0, 12, 72, 240, 600, 1260, 2352, 4032, 6480, 9900, 14520, 20592, 28392, 38220, 50400, 65280, 83232, 104652, 129960, 159600, 194040, 233772, 279312, 331200, 390000, 456300, 530712, 613872, 706440, 809100, 922560, 1047552, 1184832

Offset: 2

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002378, A002415, A006011, A008911, A083374.

[ n*(n-1)^2*(n-2): n in [2..40]]; // Vincenzo Librandi, May 02 2011

seq(floor(n^6/(n^2+1)),n=1..25); # Gary Detlefs, Feb 11 2010

f[n_]:=n*(n-1)^2*(n-2); f[Range[2,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{0,12,72,240,600},40] (* or *) CoefficientList[Series[-((12 x (1+x))/(-1+x)^5),{x,0,40}],x] (* Harvey P. Dale, Jul 31 2021 *)

a(n)=n^4 - 4*n^3 + 5*n^2 - 2*n \\ Charles R Greathouse IV, May 02 2011

a(n) = A002378(n)*A002378(n+1). - Zerinvary Lajos, Apr 11 2006
a(n) = 12*A002415(n+1) = 2*A083374(n) = 4*A006011(n+1) = 6*A008911(n+1). - Zerinvary Lajos, May 09 2007
a(n) = floor((n-1)^6/((n-1)^2+1)). - Gary Detlefs, Feb 11 2010
From Amiram Eldar, Nov 05 2020: (Start)
Sum_{n>=3} 1/a(n) = 7/4 - Pi^2/6.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/12 - 3/4. (End)
G.f.: -12*x*(1+x)/(-1+x)^5. - Harvey P. Dale, Jul 31 2021
a(n) = (n-1)^4 - (n-1)^2. - Katherine E. Stange, Mar 31 2022

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.

Original entry on oeis.org

0, 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860, 4004, 5460, 7280, 9520, 12240, 15504, 19380, 23940, 29260, 35420, 42504, 50600, 59800, 70200, 81900, 95004, 109620, 125860, 143840, 163680, 185504, 209440, 235620, 264180, 295260, 329004, 365560, 405080

Offset: 0

N. J. A. Sloane

With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - Bruno Berselli, Oct 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..700

1/beta(n, 4) in A061928.
Cf. A000332, A034827, A050534.
Cf. A002378, A005843.
Convolution of the oblong numbers with the odd numbers: A008911.
Fourth column of A003506.

[n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(4*binomial(n+3, 4), n=0..35)]; # Zerinvary Lajos, Nov 24 2006

f[n_]:=n*(n+1)*(n+2)*(n+3)/6; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
# Binomial[#+3,3]&/@ Range[0,40]  (* Harvey P. Dale, Feb 20 2011 *)

A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* Martin Ettl, Jan 22 2013 */

a(n) = n*C(3+n, 3). - Zerinvary Lajos, Jan 10 2006
G.f.: 4*x/(1-x)^5. - Colin Barker, Mar 01 2012
G.f.: (2*x/(1-x))*W(0), where W(k) = 1 + 1/( 1 - x*(k+2)*(k+4)/( x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1) ) ); (continued fraction). - Sergei N. Gladkovskii, Aug 24 2013
From Amiram Eldar, Jun 02 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2) - 16/3. (End)
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/6. - Stefano Spezia, Jul 11 2025

A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.

Original entry on oeis.org

0, 0, 8, 48, 160, 400, 840, 1568, 2688, 4320, 6600, 9680, 13728, 18928, 25480, 33600, 43520, 55488, 69768, 86640, 106400, 129360, 155848, 186208, 220800, 260000, 304200, 353808, 409248, 470960, 539400, 615040, 698368, 789888, 890120, 999600

Offset: 0

Henry Bottomley, Jul 14 2002

			a(2)=8 since for a random walk with absorbing boundaries at +2 or -2, the probability of first reaching a boundary at time t=2 is 1/2, at t=4 is 1/4, at t=6 is 1/8, at t=8 is 1/16, etc., giving a mean of 2/2 + 4/4 + 6/8 + 8/16 + ... = 4 and a variance of 2^2/2 + 4^2/4 + 6^2/8 + 8^2/16 + ... - 4^2 = 24 - 16 = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290 (i.e., n^2) for mean time. A072818(n)=sqrt(a(A001079(n))) attempts to identify the integer standard deviations.

[n^2*(n^2-1)*2/3: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

CoefficientList[Series[8 (1 + x) x^2/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, Jul 02 2019 *)

a(n) = n^2*(n^2 - 1)*2/3 = 4*A008911(n) = 8*A002415(n) = A069971(n, n).
G.f.: 8*(1 + x)*x^2/(1 - x)^5. - Arkadiusz Wesolowski, Feb 08 2012
E.g.f.: 2*exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 12 2021
a(n) = 2*n * A007290(n+1). - C.S. Elder, Jan 09 2024

A183157 Triangle read by rows: T(n,k) is the number of partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 10, 2, 1, 16, 28, 12, 2, 1, 25, 60, 40, 14, 2, 1, 36, 110, 100, 54, 16, 2, 1, 49, 182, 210, 154, 70, 18, 2, 1, 64, 280, 392, 364, 224, 88, 20, 2, 1, 81, 408, 672, 756, 588, 312, 108, 22, 2, 1, 100, 570, 1080, 1428, 1344, 900, 420, 130, 24, 2

Offset: 0

Abdullahi Umar, Dec 28 2010

Rows also give the coefficients of the clique polynomial of the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

			T (3,2) = 10 because there are exactly 10 partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,1); (1,2)-->(2,3); (1,2)-->(3,2); (2,3)-->(1,2); (2,3)-->(2,1); (2,3)-->(2,3); (2,3)-->(3,2); (1,3)-->(1,3); (1,3)-->(3,1) - the mappings are coordinate-wise.
The triangle starts
  1;
  1,    1;
  1,    4,    2;
  1,    9,   10,    2;
  1,   16,   28,   12,    2;
  1,   25,   60,   40,   14,    2;
  1,   36,  110,  100,   54,   16,    2;
  1,   49,  182,  210,  154,   70,   18,    2;
  1,   64,  280,  392,  364,  224,   88,   20,    2;
  1,   81,  408,  672,  756,  588,  312,  108,   22,    2;
  1,  100,  570, 1080, 1428, 1344,  900,  420,  130,   24,    2;

R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Clique Polynomial

Cf. A183156 (row sums), A006331 (k=2), A008911 (k=3), A067056 (k=4).

A183157 := proc(n,k) if k =0 then 1; elif k = 1 then n^2 ; else 2*(2*n-k+1)*binomial(n,k)/(k+1) ; end if; end proc: # R. J. Mathar, Jan 06 2011

T[, 0] = 1; T[n, 1] := n^2; T[n_, k_] := 2*(2*n - k + 1)*Binomial[n, k] / (k + 1);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

T(n,0)=1, T(n,1) = n^2 and T(n,k)=2*(2*n-k+1)*binomial(n,k)/(k+1), k > 1.

A348670 Decimal expansion of 10 - Pi^2.

Original entry on oeis.org

1, 3, 0, 3, 9, 5, 5, 9, 8, 9, 1, 0, 6, 4, 1, 3, 8, 1, 1, 6, 5, 5, 0, 9, 0, 0, 0, 1, 2, 3, 8, 4, 8, 8, 6, 4, 6, 8, 6, 3, 0, 0, 5, 9, 2, 7, 5, 9, 2, 0, 9, 3, 7, 3, 5, 8, 6, 6, 5, 0, 6, 2, 3, 7, 7, 9, 9, 5, 5, 1, 7, 7, 5, 8, 0, 7, 9, 4, 7, 5, 6, 9, 9, 8, 2, 2, 6, 5, 9, 6, 2, 8, 1, 4, 4, 7, 7, 6, 8, 1, 7, 5, 9, 7, 4

Offset: 0

Amiram Eldar, Oct 29 2021

Let ABC be a unit-area triangle, and let P be a point uniformly picked at random inside it. Let D, E and F be the intersection points of the lines AP, BP and CP with the sides BC, CA and AB, respectively. Then, the expected value of the area of the triangle DEF is this constant.

			0.13039559891064138116550900012384886468630059275920...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.
A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.

R. W. Gosper, Acceleration of Series, AIM-304 (1974), page 71.
Olivier Schneegans, How Close to 10 is Pi^2?, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 448.
Daniel Sitaru, Problem B131, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.
Index entries for transcendental numbers.

Cf. A002388, A008911, A010467, A060459.

Mathematica
```
RealDigits[10 - Pi^2, 10, 100][[1]]
```
PARI
```
10 - Pi^2 \\ Michel Marcus, Oct 29 2021
```

Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).
Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).
Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.
Equals 4 * Sum_{m,n>=1} (m-n)^2/(m*n*(m+1)^2*(n+1)^2*(m+2)*(n+2)) (Sitaru, 2023). - Amiram Eldar, Aug 18 2023

A256550 Triangle read by rows, T(n,k) = EL(n,k)/(n-k+1)! and EL(n,k) the matrix-exponential of the unsigned Lah numbers scaled by exp(-1), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 12, 6, 1, 0, 15, 50, 40, 10, 1, 0, 52, 225, 250, 100, 15, 1, 0, 203, 1092, 1575, 875, 210, 21, 1, 0, 877, 5684, 10192, 7350, 2450, 392, 28, 1, 0, 4140, 31572, 68208, 61152, 26460, 5880, 672, 36, 1

Offset: 0

Peter Luschny, Apr 01 2015

			Triangle starts:
1;
0,    1;
0,    1,    1;
0,    2,    3,    1;
0,    5,   12,    6,    1;
0,   15,   50,   40,   10,    1;
0,   52,  225,  250,  100,   15,   1;
0,  203, 1092, 1575,  875,  210,  21,  1;

Cf. A000110, A000217, A008911, A105479, A256551 (matrix inverse).

def T(dim) :
    M = matrix(ZZ, dim)
    for n in range(dim) :
        M[n, n] = 1
        for k in range(n) :
            M[n,k] = (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
    E = M.exp()/exp(1)
    for n in range(dim) :
        for k in range(n) :
            M[n,k] = E[n,k]/factorial(n-k+1)
    return M
T(8) # Computes the sequence as a lower triangular matrix.

T(n+1,1) = Bell(n) = A000110(n).
T(n+2,2) = C(n+2,2)*Bell(n) = A105479(n+2).
T(n+1,n) = A000217(n).
T(n+2,n) = A008911(n+1).

cp's OEIS Frontend

A002415 4-dimensional pyramidal numbers: a(n) = n^2*(n^2-1)/12.

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A083374 a(n) = n^2 * (n^2 - 1)/2.

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A006011 a(n) = n^2*(n^2 - 1)/4.

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A047659 Number of ways to place 3 nonattacking queens on an n X n board.

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A047928 a(n) = n*(n-1)^2*(n-2).

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A033488 a(n) = n*(n+1)*(n+2)*(n+3)/6.

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A047928 a(n) = n(n-1)^2(n-2).

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.