A078038 Expansion of (1-x)/(1+x-2*x^2-x^3).

1, -2, 4, -7, 13, -23, 42, -75, 136, -244, 441, -793, 1431, -2576, 4645, -8366, 15080, -27167, 48961, -88215, 158970, -286439, 516164, -930072, 1675961, -3019941, 5441791, -9805712, 17669353, -31838986, 57371980, -103380599, 186285573, -335674791, 604865338, -1089929347, 1963985232

Offset

0,2

Comments

From Johannes W. Meijer, May 29 2010: (Start)

The absolute values of the a(n) represent the number of ways White can force checkmate in exactly (n+1) moves, n>=0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6 and h6; Black Ke8, pawns b3, c7, d3, f7 and h7. (After Noam D. Elkies, see link; diagram 5).

The absolute values of the a(n) represent all paths of length n starting at the third (or fourth) node on the path graph P_6, see the Maple program.

(End)

For n>=1, abs(a(n-1)) is the number of compositions where there is no rise between every second pair of parts, starting with the second and third part; see example. Also, abs(a(n-1)) is the number of compositions of n where there is no fall between every second pair of parts, starting with the second and third part; see example. [Joerg Arndt, May 21 2013]

Links

Table of n, a(n) for n=0..36.

Noam D. Elkies, New Directions in Enumerative Chess Problems., arXiv:math/0508645 [math.CO]; 2005; The Electronic Journal of Combinatorics, 11 (2), 2004-2005. [From Johannes W. Meijer, May 29 2010]

Index entries for linear recurrences with constant coefficients, signature (-1,2,1).

Formula

a(n+3) = -a(n+2)+2*a(n+1)+a(n), a(0)=1, a(1)=-2, a(2)=4. - Wouter Meeussen, Jan 02 2005

a(n) = (-1)^n * (A006053(n+1) + A006053(n+2)). G.f. of |a(n)|: (1+x)/(x^3 - 2*x^2 - x + 1). - Ralf Stephan, Aug 19 2013

a(n) = Sum_{r=1..6} ((-2)^n*(1-(-1)^r)*cos(Pi*r/7)^n*cot(Pi*r/14)*sin(3*Pi*r/7))/7. - Herbert Kociemba, Sep 17 2020

Example

From Joerg Arndt, May 21 2013: (Start)
There are abs(a(6-1))=23 compositions of 6 where there is no rise between every second pair of parts:
          v   v    <--= no rise over these positions
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 2 1 ]
03:  [ 1 1 1 3 ]
04:  [ 1 2 1 1 1 ]
05:  [ 1 2 1 2 ]
06:  [ 1 2 2 1 ]
07:  [ 1 3 1 1 ]
08:  [ 1 3 2 ]
09:  [ 1 4 1 ]
10:  [ 1 5 ]
11:  [ 2 1 1 1 1 ]
12:  [ 2 1 1 2 ]
13:  [ 2 2 1 1 ]
14:  [ 2 2 2 ]
15:  [ 2 3 1 ]
16:  [ 2 4 ]
17:  [ 3 1 1 1 ]
18:  [ 3 2 1 ]
19:  [ 3 3 ]
20:  [ 4 1 1 ]
21:  [ 4 2 ]
22:  [ 5 1 ]
23:  [ 6 ]
There are abs(a(6-1))=23 compositions of 6 where there is no fall between every second pair of parts, starting with the second and third part:
          v   v    <--= no fall over these positions
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 3 ]
04:  [ 1 1 2 1 1 ]
05:  [ 1 1 2 2 ]
06:  [ 1 1 3 1 ]
07:  [ 1 1 4 ]
08:  [ 1 2 2 1 ]
09:  [ 1 2 3 ]
10:  [ 1 5 ]
11:  [ 2 1 1 1 1 ]
12:  [ 2 1 1 2 ]
13:  [ 2 1 2 1 ]
14:  [ 2 1 3 ]
15:  [ 2 2 2 ]
16:  [ 2 4 ]
17:  [ 3 1 1 1 ]
18:  [ 3 1 2 ]
19:  [ 3 3 ]
20:  [ 4 1 1 ]
21:  [ 4 2 ]
22:  [ 5 1 ]
23:  [ 6 ]
(End)

Maple

with(GraphTheory): G:= PathGraph(6): A:=AdjacencyMatrix(G): nmax:=36; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[3, k], k=1..6) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010

Mathematica

LinearRecurrence[{-1, 2, 1}, {1, -2, 4}, 40] (* Jean-François Alcover, Jan 08 2019 *)
a[n_]:=Sum[(-(-2)^(n+1)Cos[(Pi r)/7]^n Cot[(Pi r)/14]Sin[(3Pi r)/7])/7, {r, 1, 5, 2}]
Table[a[n], {n, 0, 40}]//Round (* Herbert Kociemba, Sep 17 2020 *)

Prog

(PARI) Vec((1-x)/(1+x-2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Crossrefs

Cf. A028495, A068911, A094790, A287381 (absolute values).

Sequence in context: A260917 A165648 A287381 * A190502 A048888 A280027

Adjacent sequences: A078035 A078036 A078037 * A078039 A078040 A078041

Keyword

sign,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved