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A030436
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Expansion of (1+x-2*x^2-x^3)/(1-4*x^2+2x^4).
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12
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1, 1, 2, 3, 6, 10, 20, 34, 68, 116, 232, 396, 792, 1352, 2704, 4616, 9232, 15760, 31520, 53808, 107616, 183712, 367424, 627232, 1254464, 2141504, 4283008, 7311552, 14623104, 24963200, 49926400, 85229696
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also (starting 3,6,...) the number of zig-zag paths from top to bottom of a rectangle of width 7 whose color is not that of the top right corner.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and h2; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3, g4 and h3.
Counts all paths of length n, n>=0, starting at the initial node on the path graph P_7, see the Maple program.
(End)
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LINKS
| Joseph Myers, BMO 2008-2009 Round 1 Problem 1 - Generalisation
Index to sequences with linear recurrences with constant coefficients, signature (0,4,0,-2)
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FORMULA
| a(0)=a(1)=1, a(2)=2, a(3)=3, a(n)=4a(n-2)-2a(n-4) [From Harvey P. Dale, May 11 2011]
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
with(GraphTheory): P:=7: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=31; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..P); od: seq(a(n), n=0..nmax);
(End)
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MATHEMATICA
| CoefficientList[Series[(1+x-2x^2-x^3)/(1-4x^2+2x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[{0, 4, 0, -2}, {1, 1, 2, 3}, 41] (* From Harvey P. Dale, May 11 2011 *)
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CROSSREFS
| a(n)=A030435(n)/2. a(2n)=A006012(n). a(2n+1)=A007052(n).
a(n) = A094718(7, n).
Cf. A024175 and A094803.
Cf. A000045, A038754, A028495, A030436, A061551 and A178381.
Sequence in context: A050291 A135452 A077027 * A030227 A180272 A061551
Adjacent sequences: A030433 A030434 A030435 * A030437 A030438 A030439
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Comment and link added and typo in cross-reference corrected by Joseph Myers (jsm(AT)polyomino.org.uk), Dec 24 2008, May 30 2010
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