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A183207 Termwise products of the natural numbers and odd integers repeated.

%I A183207 #54 Jul 07 2024 07:58:29
%S A183207 1,2,9,12,25,30,49,56,81,90,121,132,169,182,225,240,289,306,361,380,
%T A183207 441,462,529,552,625,650,729,756,841,870,961,992,1089,1122,1225,1260,
%U A183207 1369,1406,1521,1560,1681,1722,1849,1892,2025
%N A183207 Termwise products of the natural numbers and odd integers repeated.
%C A183207 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
%C A183207 The sequence gives the number of curved edges created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of vertices is A370980(n). See the attached images. - _Scott R. Shannon_, Jul 07 2024
%H A183207 Vincenzo Librandi, <a href="/A183207/b183207.txt">Table of n, a(n) for n = 1..1000</a>
%H A183207 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%H A183207 Scott R. Shannon, <a href="/A183207/a183207.jpg">Image for n = 3</a>. In this and other images the center of each circle of shown as a white dot.
%H A183207 Scott R. Shannon, <a href="/A183207/a183207_1.jpg">Image for n = 4</a>.
%H A183207 Scott R. Shannon, <a href="/A183207/a183207_2.jpg">Image for n = 5</a>.
%H A183207 Scott R. Shannon, <a href="/A183207/a183207_3.jpg">Image for n = 8</a>.
%F A183207 Termwise products of (1, 2, 3, 4, 5, 6, 7, 8...) and (1, 1, 3, 3, 5, 5, 7, 7,...).
%F A183207 From _R. J. Mathar_, Feb 12 2011: (Start)
%F A183207 G.f.: x*( -1-x-5*x^2-x^3 ) / ( (1+x)^2*(x-1)^3 ).
%F A183207 a(n) = n^2-n*(1+(-1)^n)/2. (End)
%F A183207 Sum_{n>=1} 1/a(n) = Pi^2/8 + log(2). - _Amiram Eldar_, Mar 15 2024
%F A183207 a(n) = A093005(n) + A370980(n) - 1, by Euler's formula. - _Scott R. Shannon_, Jul 07 2024
%e A183207 a(4) = 4*3 = 12.
%t A183207 a[n_] := n (2 Floor[(n - 1)/2] + 1); Array[a, 45] (* _Robert G. Wilson v_, Feb 11 2011 *)
%t A183207 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 A183207 (PARI) a(n) = n^2-n*(1+(-1)^n)/2;
%o A183207 (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 A183207 Cf. A093005, A370980.
%K A183207 nonn,easy
%O A183207 1,2
%A A183207 _Gary W. Adamson_, Feb 11 2011