A007009 -id:A007009 - cp's OEIS frontend

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

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

A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2).

Original entry on oeis.org

1, 3, 8, 16, 30, 50, 80, 120, 175, 245, 336, 448, 588, 756, 960, 1200, 1485, 1815, 2200, 2640, 3146, 3718, 4368, 5096, 5915, 6825, 7840, 8960, 10200, 11560, 13056, 14688, 16473, 18411, 20520, 22800, 25270, 27930, 30800, 33880, 37191, 40733, 44528

Offset: 0

N. J. A. Sloane

Given an irregular triangular matrix M with the triangular numbers in every column shifted down twice for columns >0, A002624 = M * [1, 2, 3, ...]. Example: row 4 of triangle M = (15, 6, 1), then (15, 6, 1) dot (1, 2, 3) = a(4) = 30 = (15 + 12 + 3). - Gary W. Adamson, Mar 02 2010
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums of A139600 are related to the sequence given above, e.g., Ze2(n) = a(n-1) - a(n-2) - a(n-3) + 4*a(n-4), with a(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011
8*a(n) + 16*a(n+1) + 16*a(n+2) is the number of ways to place 3 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly twice. Also true for the nonexistent terms for n=-1, n=-2 and n=-3 assuming that they are zeros. In graph-theory representation they thus form the corresponding open walk (Eulerian trail) with V={1,2,3} vertices and length of 2. - Antal Pinter, Dec 31 2015
a(n) is the number of partitions of n into parts with three kinds of 1 and two kinds of 2. - Joerg Arndt, Jan 18 2016

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 204
Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, Q(n) at p.8.
Antal Pinter, Software utility for enumerating positions of non-attacking and attacking chess pieces , Backtrack_V7Pro
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
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A047659, A139600, A180662.

[( (n+1)^4 +10*(n+1)^3 +32*(n+1)^2 +32*(n+1) +(6*(n+1) +15)*((n+1) mod 2) )/96 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

A002624:=-1/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation

f[n_] := Block[{m = n - 1}, (m^4 + 10m^3 + 32m^2 + 32m + (6m + 15)Mod[m, 2])/96]; Table[ f[n], {n, 2, 45}]
(* Or *) CoefficientList[ Series[1/((1 - x)^3 (1 - x^2)^2), {x, 0, 44}], x] (* Robert G. Wilson v, Feb 26 2005 *)

Vec(1/(1-x)^3/(1-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Apr 19 2012

a(n)=(n^4 + 14*n^3 + 68*n^2 + 136*n - n%2*(6*n + 21))/96 + 1 \\ Charles R Greathouse IV, Feb 18 2016

a(n-1) = ( n^4 +10*n^3 +32*n^2 +32*n +(6*n +15)*(n mod 2) )/96.
From Antal Pinter, Oct 03 2014: (Start)
a(n) = C(n + 2, 2) + 2*C(n, 2) + 3*C(n - 2, 2) + 4*C(n - 4, 2) + ...
a(n) = Sum_{i = 1..z} i*C(n + 4 - 2i, 2)  where z = (2*n + 3 + (-1)^n)/4.
a(n) = (3*(2*n + 7)*(-1)^n + 2*n^4 + 28*n^3 + 136*n^2 + 266*n + 171)/192.
(End)
a(n) = A007009(n+1) - A001752(n-1) for n>0. - Antal Pinter, Dec 27 2015
a(n) = Sum_{j=0..n+1} A006918(j). - Richard Turk, Feb 18 2016

Formula and more terms from Frank Ellermann, Mar 14 2002

A006009 Number of paraffins.

Original entry on oeis.org

4, 16, 48, 108, 216, 384, 640, 1000, 1500, 2160, 3024, 4116, 5488, 7168, 9216, 11664, 14580, 18000, 22000, 26620, 31944, 38016, 44928, 52728, 61516, 71344, 82320, 94500, 108000, 122880, 139264, 157216, 176868, 198288, 221616, 246924, 274360, 304000, 336000

Offset: 1

N. J. A. Sloane

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

4*A007009.

Cf. A005994, A005997.

a:= n-> (Matrix([[0$4,4,16,48,108]]). Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-4,-4,10,-4,-4,4,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..40); # Alois P. Heinz, Aug 13 2008

a[n_] := 1/16*(2*n^4+12*n^3+24*n^2+2*(9-(-1)^n)*n-3*(-1)^n+3); Array[a, 40] (* Jean-François Alcover, Mar 17 2014 *)

a(n) = 2*(A005994(n) + binomial(n, 4)).
G.f.: 4*x*(1-x^3) / ((1-x)^4*(1-x^2)^2). - Alois P. Heinz, Aug 13 2008
a(n) = Sum_{i=1..n} i*floor(i^2/2). - Enrique Pérez Herrero, Mar 10 2012

A330520 Sum of even integers <= n times the sum of odd integers <= n.

Original entry on oeis.org

0, 0, 2, 8, 24, 54, 108, 192, 320, 500, 750, 1080, 1512, 2058, 2744, 3584, 4608, 5832, 7290, 9000, 11000, 13310, 15972, 19008, 22464, 26364, 30758, 35672, 41160, 47250, 54000, 61440, 69632, 78608, 88434, 99144, 110808, 123462, 137180, 152000, 168000, 185220, 203742, 223608, 244904, 267674

Offset: 0

J. Stauduhar, Dec 17 2019

Number of crossings in a grid formed by drawing n parallel infinite-length lines perpendicular to the previous number of lines.

The sum of odd integers <= n is m^2 where m = round(n/2) is their number. The sum of even integers <= n is k(k+1) where k = floor(n/2) is their number. So a(n) = m^2*k(k+1), where the factor m appears three times. - M. F. Hasler, Dec 19 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A179824, A085537, A007009, A002620.

Cf. A000290 (sum of odd integers), A002378 (sum of even integers).

CoefficientList[Series[2 (x^2 + x + 1) x^2/((x + 1)^2*(1 - x)^5), {x, 0, 45}], x] (* Michael De Vlieger, Dec 22 2019 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,2,8,24,54,108},50] (* Harvey P. Dale, Dec 29 2021 *)

apply( A330520(n)=n\2*(n\2+1)*(n\/2)^2, [0..99]) \\ M. F. Hasler, Dec 19 2019

G.f.: 2*(x^2+x+1)*x^2/((x+1)^2*(1-x)^5).
a(n) = 2 * A007009(n-1) for n>1.
a(2k+i) = (k+i)^3 (k+1-i), with i = 0 or 1. - M. F. Hasler, Dec 19 2019
a(n) = A002378(floor(n/2)) * A000290(ceiling(n/2)). - Bernard Schott, Dec 22 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A006009 Number of paraffins.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330520 Sum of even integers <= n times the sum of odd integers <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula