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%I #45 Mar 15 2024 05:51:49
%S 1,2,9,12,25,30,49,56,81,90,121,132,169,182,225,240,289,306,361,380,
%T 441,462,529,552,625,650,729,756,841,870,961,992,1089,1122,1225,1260,
%U 1369,1406,1521,1560,1681,1722,1849,1892,2025
%N Termwise products of the natural numbers and odd integers repeated.
%C There is a chessboard of n^2 squares. A pawn is standing on the lower left corner of the chessboard O (0,0) and its primary goal is to reach the upper right corner of the chessboard N (n,n). The only moves allowed are diagonal shortcuts through squares. Once a square is crossed it is destroyed so that it is impossible to cross again. The secondary goal of the pawn on its way to N is to destroy as many squares as possible. a(n) is the maximum possible number of destroyed squares, provided the pawn has reached its primary goal. - _Ivan N. Ianakiev_, Feb 23 2014
%H Vincenzo Librandi, <a href="/A183207/b183207.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F Termwise products of (1, 2, 3, 4, 5, 6, 7, 8...) and (1, 1, 3, 3, 5, 5, 7, 7,...).
%F From _R. J. Mathar_, Feb 12 2011: (Start)
%F G.f.: x*( -1-x-5*x^2-x^3 ) / ( (1+x)^2*(x-1)^3 ).
%F a(n) = n^2-n*(1+(-1)^n)/2. (End)
%F Sum_{n>=1} 1/a(n) = Pi^2/8 + log(2). - _Amiram Eldar_, Mar 15 2024
%e a(4) = 4*3 = 12.
%t a[n_] := n (2 Floor[(n - 1)/2] + 1); Array[a, 45] (* _Robert G. Wilson v_, Feb 11 2011 *)
%t CoefficientList[Series[(-1 - x - 5 x^2 - x^3)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Feb 23 2014 *)
%o (PARI) a(n) = n^2-n*(1+(-1)^n)/2;
%o (Magma) I:=[1,2,9,12,25]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // _Vincenzo Librandi_, Feb 23 2014
%Y Cf. A093005.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Feb 11 2011
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