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

2,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A130516.

Sequence in context: A060509 A113203 A034486 * A302851 A276083 A161893

Adjacent sequences: A130512 A130513 A130514 * A130516 A130517 A130518

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 09 2007

Status

approved