A117937 -id:A117937 - cp's OEIS frontend

A061989 Number of ways to place 3 nonattacking queens on a 3 X n board.

0, 0, 0, 0, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 29 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
E. Lucas, Recreations mathematiques I, Albert Blanchard, Paris, 1992, p. 231.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A061990, A117937, A229183.

Essentially the same as A079908.

[0,0,0] cat [(n-3)*(n^2-6*n+12): n in [3..50]]; // G. C. Greubel, Apr 29 2022

A061989 := proc(n)
    if n >= 3 then
        (n-3)*(n^2-6*n+12) ;
    else
        0;
    end if;
end proc:
seq(A061989(n),n=0..30) ; # R. J. Mathar, Aug 16 2019

CoefficientList[Series[2*x^4*(2-x+2*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,0]+[(n-3)*((n-3)^2 +3) for n in (3..50)] # G. C. Greubel, Apr 29 2022

G.f.: 2*x^4*(2-x+2*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 7.
Explicit formula (H. Tarry, 1890): a(n) = (n-3)*(n^2-6*n+12), n >= 3.
(4, 14, 36, ...) is the binomial transform of row 4 of A117937: (4, 10, 12, 6). - Gary W. Adamson, Apr 09 2006
a(n) = 2*A229183(n-3). - R. J. Mathar, Aug 16 2019
E.g.f.: 36 + 14*x + 2*x^2 + (-36 + 22*x - 6*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 29 2022

A117938 Triangle, columns generated from Lucas Polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123

Offset: 1

Gary W. Adamson, Apr 03 2006

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).

A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  6,   4;
  1, 4, 11,  14,   7;
  1, 5, 18,  36,  34,  11;
  1, 6, 27,  76, 119,  82,  18;
  1, 7, 38, 140, 322, 393, 198, 29;
  ...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.

Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).

Lucas := proc(n,x) # see A114525
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
    expand(%) ;
end proc:
A117938 := proc(n::integer,k::integer)
    if k = 1 then
        1;
    else
        subs(x=n-k+1,Lucas(k-1,x)) ;
    end if;
end proc:
seq(seq(A117938(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 16 2019

T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)

def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021

Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...

Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021

A117936 Triangle, rows = inverse binomial transforms of A073133 columns.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 9, 12, 6, 5, 24, 56, 60, 24, 8, 62, 228, 414, 360, 120, 13, 156, 864, 2400, 3480, 2520, 720, 21, 387, 3132, 12606, 27360, 32640, 20160, 5040, 34, 951, 11034, 62220, 190704, 335160, 337680, 181440, 40320, 55, 2323, 38136, 294588, 1229760, 2997120, 4394880, 3820320, 1814400, 362880

Offset: 1

Gary W. Adamson, Apr 03 2006

Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.

Note that binomial transforms are defined from offset 1 here. - R. J. Mathar, Aug 16 2019

			First few columns of A073133 are: (1, 1, 1, ...); (1, 2, 3, ...); (2, 5, 10, 17, ...); (3, 12, 33, 72, ...). As sequences, these are f(x), Fibonacci polynomials: (1); (x); (x^2 + 1); (x^3 + 2*x); (x^4 + 3*x^2 + 1); (x^5 + 4*x^3 + 3*x); ... For example, f(x), x = 1,2,3,... using (x^4 + 3*x^2 + 1) generates Column 5 of A073133: (5, 29, 109, 305, ...).
Inverse binomial transforms of the foregoing columns generates the triangle rows:
  1;
  1,  1;
  2,  3,   2;
  3,  9,  12,   6;
  5, 24,  56,  60,  24;
  8, 62, 228, 414, 360, 120;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flatten

Cf. A073133, A117937, A117938.

Cf. A006684 (column 2), A309717 (column 3 halved).

A117936 := proc(n,k)
    add( A073133(i+1,n)*binomial(k-1,i)*(-1)^(i-k-1),i=0..k-1) ;
end proc:
seq(seq(A117936(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Aug 16 2019

(* A = A073133 *) A[, 1] = 1; A[n, k_] := A[n, k] = If[k < 0, 0, n A[n, k - 1] + A[n, k - 2]];
T[n_, k_] := Sum[A[i+1, n] Binomial[k-1, i] (-1)^(i - k - 1), {i, 0, k-1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)

@CachedFunction
def A073133(n,k): return 0 if (k<0) else 1 if (k==1) else n*A073133(n,k-1) + A073133(n,k-2)
def A117936(n,k): return sum( (-1)^(j-k+1)*binomial(k-1, j)*A073133(j+1,n) for j in (0..k-1) )
flatten([[A117936(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.

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A061989 Number of ways to place 3 nonattacking queens on a 3 X n board.

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A117938 Triangle, columns generated from Lucas Polynomials.

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A117936 Triangle, rows = inverse binomial transforms of A073133 columns.

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Crossrefs

Programs

Maple

Mathematica

Sage

Formula