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A078038
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Expansion of (1-x)/(1+x-2*x^2-x^3).
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4
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), 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)
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LINKS
| Noam D. Elkies, New Directions in Enumerative Chess Problems., The Electronic Journal of Combinatorics, 11 (2), 2004-2005. [From Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010]
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FORMULA
| a(n+3)=-a(n+2)+2*a(n+1)+a(n), a(0)=1, a(1)=-2, a(2)=4. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jan 02 2005
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
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);
(End)
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CROSSREFS
| Cf. A028495, A068911 and A094790.
Sequence in context: A136299 A003116 A165648 * A190502 A048888 A026724
Adjacent sequences: A078035 A078036 A078037 * A078039 A078040 A078041
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002
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