A163102 -id:A163102 - cp's OEIS frontend

A317714 Chessboard rectangles sequence (see Comments), also A037270 interleaved with A163102.

0, 0, 1, 2, 10, 18, 45, 72, 136, 200, 325, 450, 666, 882, 1225, 1568, 2080, 2592, 3321, 4050, 5050, 6050, 7381, 8712, 10440, 12168, 14365, 16562, 19306, 22050, 25425, 28800, 32896, 36992, 41905, 46818, 52650, 58482, 65341, 72200, 80200, 88200, 97461, 106722, 117370

Offset: 1

Ivan N. Ianakiev, Aug 05 2018

Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite rectangles of p X q unit squares whose vertices are covered by black unit squares (1 < p <= n, 1 < q <= n).

			In a 4 X 4 chessboard there are two such rectangles (for both p = q = 3) and the coordinates of their lower left vertices are a1 and b2. Therefore, a(4) = 2.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A005993, A037270, A163102.

[(5-5*(-1)^n-12*n+12*(-1)^n*n+14*n^2-6*(-1)^n*n^2-8*n^3+2*n^4)/64: n in [1..50]]; // Vincenzo Librandi, Aug 05 2018

CoefficientList[Series[-((x^2 (1+4 x^2+x^4))/((-1+x)^5 (1+x)^3)),{x,0,44}],x]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 1, 2, 10, 18, 45, 72}, 80] (* Vincenzo Librandi, Aug 06 2018 *)

a(n) = sum(i = 1, n-1, floor(i/2)^3); \\ Jinyuan Wang, Aug 12 2019

n, a = 0, 0
while n < 10:
    print(n,a)
n, a = n+1, a+((n+1)//2)**3 # A.H.M. Smeets, Aug 09 2019

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8), with a(1)=0, a(2)=0, a(3)=1, a(4)=2, a(5)=10, a(6)=18, a(7)=45, a(8)=72.
G.f.: -(x^3*(1+ 4*x^2 + x^4))/((-1+x)^5*(1+x)^3).
a(n) = (5 - 5*(-1)^n - 12*n + 12*(-1)^n*n + 14*n^2 - 6*(-1)^n*n^2 - 8*n^3 + 2*n^4)/64.
a(n) = Sum_{i=1..n-1} floor(i/2)^3. - Ridouane Oudra, Jul 24 2019
E.g.f.: (1/64)*exp(-x)*(-5-6*x-6*x^2+exp(2*x)*(5-4*x+4*x^2+4*x^3+2*x^4)). - Stefano Spezia, Aug 14 2019
a(2*n) = A163102(n-1) and a(2*n+1) = A037270(n). - Ridouane Oudra, Mar 24 2024
Sum_{n>=3} 1/a(n) = Pi^2 - Pi*coth(Pi) - 5. - Amiram Eldar, Jul 04 2025

A006331 a(n) = n(n+1)(2*n+1)/3.

Original entry on oeis.org

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

N. J. A. Sloane

Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020
a(n) is the irregularity of the n-th power of a path of length at least 3*n.  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023
a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012

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
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Rowan Beckworth, Basic atomic information.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
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
Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A132339.
Cf. A002378, A048147, A098077, A163102.
Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI
```
a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
```

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
From Bruno Berselli, May 17 2018: (Start)
a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018
E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022

A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard.

Original entry on oeis.org

0, 4, 36, 144, 400, 900, 1764, 3136, 5184, 8100, 12100, 17424, 24336, 33124, 44100, 57600, 73984, 93636, 116964, 144400, 176400, 213444, 256036, 304704, 360000, 422500, 492804, 571536, 659344, 756900, 864900, 984064, 1115136, 1258884

Offset: 1

Erich Friedman

a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for fixed different x_1, x_2 in {1,2,3,4} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 17 2007
The third differences of certain values of the hypergeometric function 3F2 lead to this sequence, i.e., 3F2([1,n+1,n+1], [n+2,n+2], z=1) - 3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,n+3], [n+4,n+4], z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 with n = -1, 0, 1, 2, ... See also A162990. - Johannes W. Meijer, Jul 21 2009
a(n) is the denominator (m*n)^2 of the term (1/m^2 - 1/n^2) = (2*n-1)/(m*n)^2, n = m+1, m > 0 in the Rydberg formula, while A005408 is the numerator 2n-1. So the quotient A005408/A035287 simulates the hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Leo Tavares, Illustration: Square of squares
Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site. [From _Johannes W. Meijer_, Jul 21 2009]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002378.

Cf. A000290.

[n^2 * (n-1)^2: n in [1..40]]; // Vincenzo Librandi, May 21 2011

Table[(n - 1)^2 n^2, {n, 30}] (* Alonso del Arte, May 20 2011 *)

for n in range(100):
  print(((n+1)*n)**2) # John H. Chakkour, Dec 14 2019

[n^2*(n-1)^2 for n in range(1, 35)] # Zerinvary Lajos, Dec 03 2009

a(n) = n^2 * (n-1)^2.
a(n) = A002378(n-1)^2. - Zerinvary Lajos, Apr 11 2006
From Stephen Crowley, Jul 19 2009: (Start)
a(n) = n!*(2*n+1) / lim_{x->0} (d^n/dx^n) (polylog(2,x)*(1-1/x));
Sum_{n >= 2} 1/a(n) = 2*zeta(2) - 3 = A145426. (End) [Comment from Jianing Song, Dec 31 2022: Note that polylog(2,x)*(1-1/x) = -1 + Sum_{n>=1} ((2*n+1)/(n^2*(n+1)^2))*x^n, so (d^n/dx^n) (polylog(2,x)*(1-1/x)) = n!*(2*n+1)/(n^2*(n+1)^2) for n >= 1. - Jianing Song, Dec 31 2022]
a(n) = 4*A000537(n-1) = 2*A163102(n-1). - Omar E. Pol, Nov 29 2011
G.f.: 4*x^2*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 04 2012
a(n) = 4*A000217(n-1)^2. - J. M. Bergot, Nov 01 2012
E.g.f.: x^2*(2 + 4*x + x^2)*exp(x). - Ilya Gutkovskiy, May 24 2016
Sum_{n>=2} (-1)^n/a(n) = 3 - 4*log(2). - Amiram Eldar, Jul 02 2020
Product_{n>=2} (1 - 1/a(n)) = -cos(sqrt(5)*Pi/2)*cosh(sqrt(3)*Pi/2)/Pi^2. - Amiram Eldar, Jan 29 2021
(n^2)^2 + (n^2+1)^2 + ... + (n^2 + n)^2 + a(n) = (n^2 + n + 1)^2 + ... + (n^2 + 2*n)^2. - Charlie Marion, Jun 18 2022
a(n) = A000290(n-1) * A000290(n). - Leo Tavares, Dec 03 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n >= 6. - Jianing Song, Dec 30 2022

A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320

Offset: 0

Abdullahi Umar, Sep 10 2008, Sep 30 2008

T(n,k) is also the number of elements in the Green's J equivalence classes in the symmetric inverse monoid, I sub n.
T(n,k) is also the number of ways to place k nonattacking rooks on an n X n chessboard. It can be obtained by performing P(n,k) permutations of n-columns over each C(n,k) combination of n-rows for the given k-rooks. The rule is also applicable for unequal (m X n) sized rectangular boards. - Antal Pinter, Nov 12 2014
Rows also give the coefficients of the matching-generating polynomial of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Rows also give the coefficients of the independence polynomial of the n X n rook graph and clique polynomial of the n X n rook complement graph. - Eric W. Weisstein, Jun 13 and Sep 14 2017
T(n,k) is the number of increasing subsequences of length n-k over all permutations of [n]. - Geoffrey Critzer, Jan 08 2023

			T(3,1) = 9 because there are exactly 9 partial bijections (on a 3-element set) of height 1, namely: (1)->(1), (1)->(2), (1)->(3), (2)->(1), (2)->(2), (2)->(3), (3)->(1), (3)->(2), (3)->(3).
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   2;
  1,  9,  18,    6;
  1, 16,  72,   96,   24;
  1, 25, 200,  600,  600,  120;
  1, 36, 450, 2400, 5400, 4320, 720;
  ...

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.
J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995).
Vaclav Kotesovec, Non-attacking chess pieces, 6th ed. (2013), p. 216, p. 218.

Alois P. Heinz, Rows n = 0..140, flattened
Wayne A. Johnson, Exponential Hilbert series of equivariant embeddings, arXiv:1804.04943 [math.RT], 2018.
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18.
Eric Weisstein's World of Mathematics, Clique Polynomial
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph

T(n,k) = |A021010|. Sum of rows of T(n,k) is A002720. T(n,n) is the order of the symmetric group on an n-element set, n!.
Columns k=2..10 are A163102, A179058, A179059, A179060, A179061, A179062, A179063, A179064, A179065.
Cf. A089231, A105219, A295383.

/* As triangle */ [[(Binomial(n,k)^2)*Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jun 13 2017

T:= (n, k)-> (binomial(n, k)^2)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 04 2012

Table[Table[Binomial[n, k]^2 k!,{k, 0, n}], {n, 0, 6}] // Flatten (* Geoffrey Critzer, Dec 04 2012 *)
Table[ CoefficientList[n!*LaguerreL[n, x], x] // Abs // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 18 2013 *)
CoefficientList[Table[n! x^n LaguerreL[n, -1/x], {n, 0, 8}], x] // Flatten (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Table[(-x)^n HypergeometricU[-n, 1, -(1/x)], {n, 5}],
  x] // Flatten (* Eric W. Weisstein, Jun 13 2017 *)

T(n,k) = k! * binomial(n,k)^2 \\ Andrew Howroyd, Feb 13 2018

T(n,k) = (C(n,k)^2)*k!.
T(n,k) = A007318(n,k) * A008279(n,k). - Antal Pinter, Nov 12 2014
From Peter Bala, Jul 04 2016: (Start)
G.f.: exp(x*t)*I_0(2*sqrt(x)) = 1 + (1 + t)*x/1!^2 + (1 + 4*t + 2*t^2)*x^2/2!^2 + (1 + 9*t + 18*t^2 + 6*t^3)*x^3/3!^2 + ..., where I_0(x) = Sum_{n >= 0} (x/2)^(2*n)/n!^2 is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*n!/k!*binomial(n,k) *t^(n-k)*R(k,t). Cf. A089231. (End)
From Peter Bala, Oct 05 2019: (Start)
E.g.f.: 1/(1 - t*x)*exp(x/(1 - t*x)).
Recurrence for row polynomials: R(n+1,t) = (1 + (2*n+1)*t)R(n,t) - n^2*t^2*R(n-1,t), with R(0,t) = 1 and R(1,t) = 1 + t.
R(n,t) equals the denominator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The numerator polynomial is the (n+1)-th row polynomial of A089231. (End)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/A001044(n) = exp(y*x)*E(x) where E(x) = Sum_{n>=0} x^n/A001044(n). - Geoffrey Critzer, Jan 08 2023
Sum_{k=0..n} k*T(n,k) = A105219(n). - Alois P. Heinz, Jan 08 2023
T(n,k) = Sum_{d=0..2*k} c(k,d)*n^d, where c(k,d) = Sum_{j=max(d-k,0)..k} binomial(k,j)*A008275(k+j,d)/j!. - Eder G. Santos, Jan 23 2025

A036464 Number of ways to place two nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 8, 44, 140, 340, 700, 1288, 2184, 3480, 5280, 7700, 10868, 14924, 20020, 26320, 34000, 43248, 54264, 67260, 82460, 100100, 120428, 143704, 170200, 200200, 234000, 271908, 314244, 361340, 413540, 471200, 534688, 604384

Offset: 1

Robert G. Wilson v, Raymond Bush (c17h21no4(AT)hotmail.com), Kirk Conely, N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, January 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.
V. Kotesovec, Non-attacking chess pieces
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A047659, A061994, A108792, A176186, A178721.

Column k=2 of A348129.

f:=n->n^4/2 - 5*n^3/3 + 3*n^2/2 - n/3; [seq(f(n),n=1..200)]; # N. J. A. Sloane, Feb 16 2013

f[k_] := 2 k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]   (* A036464 *)
Table[a[n]/4, {n, 2, 50}] (* A000914 *)
(* Clark Kimberling, Dec 31 2011 *)
CoefficientList[Series[4 x^2 (2 + x) / (1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,8,44,140},50] (* Harvey P. Dale, Mar 26 2015 *)

a(n) = C(n, 3)*(3*n-1).
G.f.: 4*x^3*(2+x)/(1-x)^5. - Colin Barker, May 02 2012
a(n) = 2*sum_{i=1..n-2} i(i + 1)^2. - Wesley Ivan Hurt, Mar 18 2014
E.g.f.: (exp(x) * x^3 * (8 + 3*x))/6. - Vaclav Kotesovec, Feb 15 2015
For n>0, a(n) = A163102(n-1) - A006331(n-1). - Antal Pinter, Sep 20 2015

A323541 a(n) = Product_{k=0..n} (k^3 + (n-k)^3).

Original entry on oeis.org

0, 1, 128, 59049, 51380224, 80869140625, 207351578198016, 811509810302822449, 4603095542875667038208, 36344623587588604291790241, 386644580358400000000000000000, 5395532942025804980378907333844441, 96578621213529440721046520779140759552

Offset: 0

Vaclav Kotesovec, Jan 17 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..158

Cf. A272246, A323496, A323540, A323542, A323543, A323544, A323545, A323546.

Cf. 2*A000537 and A163102 (with sum instead of product).

m:=3; [(&*[k^m + (n-k)^m: k in [0..n]]): n in [0..15]]; // G. C. Greubel, Jan 18 2019

Table[Product[k^3+(n-k)^3, {k, 0, n}], {n, 0, 15}]

m=3; vector(15, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019

m=3; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..15)] # G. C. Greubel, Jan 18 2019

a(n) ~ exp(2*(Pi/(3*sqrt(3))-1)*n) * n^(3*n + 3).

A060300 a(n) = (2n(n+1))^2.

Original entry on oeis.org

0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600, 3459600, 3936256, 4460544, 5035536, 5664400

Offset: 0

Jason Earls, Mar 25 2001

Arises from middle column 4^2, 12^2, 24^2, ... of following triangle: :
                        3^2 +  4^2 = 5^2
                10^2 + 11^2 + 12^2 = 13^2 + 14^2
         21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
  36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2
...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000537, A002378, A035287, A046092, A163102, A254371.

[(2*n*(n+1))^2: n in [0..30]]; // Vincenzo Librandi, Nov 18 2016

CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Nov 18 2016 *)
Table[(2n(n+1))^2,{n,0,30}] (* Harvey P. Dale, Jan 19 2019 *)

a(n) = { (2*n*(n + 1))^2 } \\ Harry J. Smith, Jul 03 2009

G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 22 2012
a(n) = 4*A035287(n+1) = 4*A002378(n)^2. - Michel Marcus, May 24 2016
a(n) = 16*A000537(n) = 16*(n*(n+1)/2)^2 = 16*A000217(n)^2 = A046092(n)^2. - Bruce J. Nicholson, Jun 05 2017
a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - César Aguilera, Jun 27 2020
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 4*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 2*A254371(n) = 8*A163102(n). (End)

Name corrected by Harry J. Smith, Jul 03 2009

A179058 Number of non-attacking placements of 3 rooks on an n X n board.

Original entry on oeis.org

0, 0, 6, 96, 600, 2400, 7350, 18816, 42336, 86400, 163350, 290400, 490776, 794976, 1242150, 1881600, 2774400, 3995136, 5633766, 7797600, 10613400, 14229600, 18818646, 24579456, 31740000, 40560000, 51333750, 64393056, 80110296

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Also the number of 3-cycles in the n X n rook complement graph. - Eric W. Weisstein, Sep 05 2017
Also the number of 6-cycles in the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Dec 07 2023
Essentially the same as A303212. - Eric W. Weisstein, Dec 06 2023

Andrew Howroyd, Table of n, a(n) for n = 1..200
Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Column k=3 of A144084.
Cf. A163102 (2 rooks), A179059 (4 rooks).
Cf. A291910 (4-cycles), A291911 (5-cycles), A291912 (6-cycles).

(* Start from Eric W. Weisstein, Sep 05 2017 *)
Table[3! Binomial[n, 3]^2, {n, 20}]
3! Binomial[Range[20], 3]^2
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, 20]
CoefficientList[Series[-((6 x^2 (1 + 9 x + 9 x^2 + x^3))/(-1 + x)^7), {x, 0, 20}], x]
(* End *)
a[n_] := If[n<3, 0, Coefficient[n!*LaguerreL[n, x], x, n-3] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

a(n) = 3!*binomial(n, 3)^2; \\ Andrew Howroyd, Feb 13 2018

a(n) = 3!*binomial(n, 3)^2.
a(n) = (n^2*(2-3*n+n^2)^2)/6. - Colin Barker, Jan 08 2013
G.f.: -6*x^3*(x+1)*(x^2+8*x+1) / (x-1)^7. - Colin Barker, Jan 08 2013
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Eric W. Weisstein, Sep 05 2017
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=3} 1/a(n) = 3*Pi^2/2 - 117/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = 21/8 - Pi^2/4. (End)

A163274 a(n) = n^4*(n+1)^2/2.

Original entry on oeis.org

0, 2, 72, 648, 3200, 11250, 31752, 76832, 165888, 328050, 605000, 1054152, 1752192, 2798978, 4321800, 6480000, 9469952, 13530402, 18948168, 26064200, 35280000, 47064402, 61960712, 80594208, 103680000, 132031250, 166567752, 208324872

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163284.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006002, A099903, A163102, A163275, A163276, A163277, A163284.

Table[(n^4 (n+1)^2)/2,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,2,72,648,3200,11250,31752},30] (* Harvey P. Dale, May 07 2012 *)

a(n)=n^4*(n+1)^2/2 \\ Charles R Greathouse IV, Oct 07 2015

From R. J. Mathar, Jul 29 2009: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: -2*x*(1 + 29*x + 93*x^2 + 53*x^3 + 4*x^4)/(x-1)^7. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi^2/3 + Pi^4/45 - 4*zeta(3) - 10.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10 + Pi^2/3 + 7*Pi^4/360 - 16*log(2) - 3*zeta(3). (End)

More terms from R. J. Mathar, Jul 29 2009

A163275 a(n) = n^5*(n+1)^2/2.

Original entry on oeis.org

0, 2, 144, 1944, 12800, 56250, 190512, 537824, 1327104, 2952450, 6050000, 11595672, 21026304, 36386714, 60505200, 97200000, 151519232, 230016834, 341067024, 495219800, 705600000, 988352442, 1363135664, 1853666784

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163285.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A006002, A099903, A163102, A163274, A163276, A163277, A163285.

A163275 := proc(n) n^5*(n+1)^2/2 ; end proc: seq(A163275(n),n=0..60) ; # R. J. Mathar, Feb 05 2010

Table[(1/2)*n^5*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{8,-28,56, -70,56,-28,8,-1}, {0,2,144,1944,12800,56250,190512,537824}, 50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1+64*x+424*x^2+584*x^3+179*x^4+8*x^5)/(x-1)^8 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: 2*x*(1 + 64*x + 424*x^2 + 584*x^3 + 179*x^4  +8*x^5)/(x-1)^8. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 12 -5*Pi^2/3 - 2*Pi^4/45 + 6*zeta(3) + 2*zeta(5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20*log(2) + 9*zeta(3)/2 + 15*zeta(5)/8 - 12 - Pi^2/2 - 7*Pi^4/180. (End)

Extended by R. J. Mathar, Feb 05 2010

cp's OEIS Frontend

A317714 Chessboard rectangles sequence (see Comments), also A037270 interleaved with A163102.

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A006331 a(n) = n*(n+1)*(2*n+1)/3.

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A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard.

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A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

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A036464 Number of ways to place two nonattacking queens on an n X n board.

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A323541 a(n) = Product_{k=0..n} (k^3 + (n-k)^3).

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A006331 a(n) = n(n+1)(2*n+1)/3.

A060300 a(n) = (2n(n+1))^2.