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A078038 Expansion of (1-x)/(1+x-2*x^2-x^3). 7
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 (list; graph; refs; listen; history; text; internal format)
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., 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

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:

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:

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-Fran├žois Alcover, Jan 08 2019 *)

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.

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

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Last modified August 14 01:55 EDT 2020. Contains 336476 sequences. (Running on oeis4.)