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A130515 In triangular peg solitaire, number of distinct feasible pairs starting with one peg missing and finishing with one peg.

%I A130515 #19 Nov 26 2017 09:49:29
%S A130515 1,4,3,17,29,27,80,125,108,260,356,300,637,832,675,1341,1665,1323,
%T A130515 2500,3025,2352,4304,5072,3888,6929,8036,6075,10625,12125,9075,15616,
%U A130515 17629,13068,22212,24804,18252,30685,34000,24843,41405,45521
%N A130515 In triangular peg solitaire, number of distinct feasible pairs starting with one peg missing and finishing with one peg.
%C A130515 Coincides with A130516 for n >= 6.
%H A130515 George I. Bell, <a href="/A130515/b130515.txt">Table of n, a(n) for n = 2..52</a>
%H A130515 George I. Bell, <a href="https://arxiv.org/abs/math/0703865">Solving Triangular Peg Solitaire</a>, arXiv:math/0703865 [math.CO], 2007-2009.
%H A130515 G. I. Bell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Bell/bell2.html">Solving Triangular Peg Solitaire</a>, JIS 11 (2008) 08.4.8
%H A130515 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 3, 0, 0, -1, 0, 0, -5, 0, 0, 5, 0, 0, 1, 0, 0, -3, 0, 0, 1).
%F A130515 Reference gives an explicit formula for a(n).
%F A130515 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
%F A130515 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
%p A130515 A130515 := proc(n)
%p A130515     t := n*(n+1)/2 ;
%p A130515     if modp(n,3) = 1 then
%p A130515         (t-1)^2/27 ;
%p A130515     elif type(n,'even') then
%p A130515         (4*t^2+9*n^2)/72 ;
%p A130515     else
%p A130515         (4*t^2+9*(n+1)^2)/72 ;
%p A130515     fi;
%p A130515 end proc: # _R. J. Mathar_, Sep 07 2015
%t A130515 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]];
%t A130515 Table[a[n], {n, 2, 42}] (* _Jean-François Alcover_, Nov 26 2017 *)
%o A130515 (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
%Y A130515 Cf. A130516.
%K A130515 nonn,easy
%O A130515 2,2
%A A130515 _N. J. A. Sloane_, Aug 09 2007