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

1, 2, 9, 12, 25, 30, 49, 56, 81, 90, 121, 132, 169, 182, 225, 240, 289, 306, 361, 380, 441, 462, 529, 552, 625, 650, 729, 756, 841, 870, 961, 992, 1089, 1122, 1225, 1260, 1369, 1406, 1521, 1560, 1681, 1722, 1849, 1892, 2025

Offset: 1

Author

Gary W. Adamson, Feb 11 2011

Keywords

• nonn
• easy

Comments

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

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

Examples

			a(4) = 4*3 = 12.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
• Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 8.

Crossrefs

Cf. A093005, A370980.

Programs

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

Mathematica

a[n_] := n (2 Floor[(n - 1)/2] + 1); Array[a, 45] (* Robert G. Wilson v, Feb 11 2011 *)
CoefficientList[Series[(-1 - x - 5 x^2 - x^3)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 23 2014 *)

PARI

a(n) = n^2-n*(1+(-1)^n)/2;

Formula

Termwise products of (1, 2, 3, 4, 5, 6, 7, 8...) and (1, 1, 3, 3, 5, 5, 7, 7,...).

From R. J. Mathar, Feb 12 2011: (Start)

G.f.: x*( -1-x-5*x^2-x^3 ) / ( (1+x)^2*(x-1)^3 ).

a(n) = n^2-n*(1+(-1)^n)/2. (End)

Sum_{n>=1} 1/a(n) = Pi^2/8 + log(2). - Amiram Eldar, Mar 15 2024

a(n) = A093005(n) + A370980(n) - 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024