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

Author

N. J. A. Sloane, Aug 09 2007

Keywords

• nonn
• easy

Comments

Coincides with A130516 for n >= 6.

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

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

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