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

1,2

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

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).

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

Example

a(4) = 4*3 = 12.

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 *)

Prog

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

Crossrefs

Cf. A093005.

Sequence in context: A129829 A053900 A318681 * A347496 A304797 A337360

Adjacent sequences: A183204 A183205 A183206 * A183208 A183209 A183210

Keyword

nonn,easy

Author

Gary W. Adamson, Feb 11 2011

Status

approved