A130515 In triangular peg solitaire, number of distinct feasible pairs starting with one peg missing and finishing with one peg.
1, 4, 3, 17, 29, 27, 80, 125, 108, 260, 356, 300, 637, 832, 675, 1341, 1665, 1323, 2500, 3025, 2352, 4304, 5072, 3888, 6929, 8036, 6075, 10625, 12125, 9075, 15616, 17629, 13068, 22212, 24804, 18252, 30685, 34000, 24843, 41405, 45521
Offset: 2
Links
- George I. Bell, Table of n, a(n) for n = 2..52
- George I. Bell, Solving Triangular Peg Solitaire, arXiv:math/0703865 [math.CO], 2007-2009.
- G. I. Bell, Solving Triangular Peg Solitaire, JIS 11 (2008) 08.4.8
- Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -1, 0, 0, -5, 0, 0, 5, 0, 0, 1, 0, 0, -3, 0, 0, 1).
Crossrefs
Cf. A130516.
Programs
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Maple
A130515 := proc(n) t := n*(n+1)/2 ; if modp(n,3) = 1 then (t-1)^2/27 ; elif type(n,'even') then (4*t^2+9*n^2)/72 ; else (4*t^2+9*(n+1)^2)/72 ; fi; end proc: # R. J. Mathar, Sep 07 2015
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Mathematica
a[n_] := With[{t = n*(n + 1)/2}, Which[Mod[n, 3] == 1, (t - 1)^2/27, EvenQ[n], (4*t^2 + 9*n^2)/72, True, (4*t^2 + 9*(n + 1)^2)/72]]; Table[a[n], {n, 2, 42}] (* Jean-François Alcover, Nov 26 2017 *) -
PARI
a(n) = {my(T = n*(n+1)/2); if (n % 3 == 1, (T-1)^2/27, if ( n % 2 == 0, (4*T^2 + 9*n^2)/72, (4*T^2 + 9*(n+1)^2)/72;););} \\ Michel Marcus, Apr 21 2013
Formula
Reference gives an explicit formula for a(n).
G.f.: -x^2*(x^2+1) *(x^14 +4*x^13 +2*x^12 +10*x^11 +15*x^10 +8*x^9 +15*x^8 +34*x^7 +15*x^6 +8*x^5 +15*x^4 +10*x^3 +2*x^2 +4*x +1) / ( (1+x)^2 *(x^2-x+1)^2 *(x-1)^5 *(1+x+x^2)^5 ). - R. J. Mathar, Sep 07 2015
a(n) = 3*a(n-3) -a(n-6) -5*a(n-9) +5*a(n-12) +a(n-15) -3*a(n-18) +a(n-21). - R. J. Mathar, Sep 07 2015
Comments