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A178381
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Number of paths of length n starting at initial node of the path graph P_9.
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18
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1, 1, 2, 3, 6, 10, 20, 35, 70, 125, 250, 450, 900, 1625, 3250, 5875, 11750, 21250, 42500, 76875, 153750, 278125, 556250, 1006250, 2012500, 3640625, 7281250, 13171875, 26343750, 47656250, 95312500, 172421875, 344843750
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OFFSET
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0,3
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COMMENTS
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Counts all paths of length n, n>=0, starting at initial node on the path graph P_9, see the Maple program.
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 g4; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3 and g5.
The path graphs P_(2*p) have as limit(a(n+1)/a(n), n =infinity) = 2 resp. hypergeom([(p-1)/(2*p+1),(p+2)/(2*p+1)],[1/2],3/4) and the path graphs P_(2*p+1) have as limit(a(n+1)/a(n), n =infinity) = 1+cos(Pi/(p+1)), p>=1; see the crossrefs. - Johannes W. Meijer, Jul 01 2010
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LINKS
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FORMULA
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G.f.: (1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=5 with a(0)=1, a(1)=1, a(2)=2, a(3)=3 and a(4)=6.
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x / (1 - x / (1 + x / (1 + x)))))))). - Michael Somos, Feb 08 2015
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 70*x^8 + ...
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MAPLE
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with(GraphTheory): P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=36; 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);
r := j -> (-1)^(j/10) - (-1)^(1-j/10):
a := k -> add((2 + r(j))*r(j)^k, j in [1, 3, 5, 7, 9])/10:
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MATHEMATICA
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CoefficientList[Series[(1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4), {x, 0, 50}], x] (* G. C. Greubel, Sep 18 2018 *)
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PROG
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(PARI) x='x+O('x^50); Vec((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4)) \\ G. C. Greubel, Sep 18 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4))); // G. C. Greubel, Sep 18 2018
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CROSSREFS
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Cf. A000007 (P_1), A000012 (P_2), A016116 (P_3), A000045 (P_4), A038754 (P_5), A028495 (P_6), A030436 (P_7), A061551 (P_8), this sequence (P_9), A336675 (P_10), A336678 (P_11), and A001405 (P_infinity).
Cf. A216212 (P_9 starting in the middle).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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