A059924 - OEIS

%I #28 Dec 30 2024 22:43:30

%S 0,2,10,34,80,158,274,438,656,938,1290,1722,2240,2854,3570,4398,5344,

%T 6418,7626,8978,10480,12142,13970,15974,18160,20538,23114,25898,28896,

%U 32118,35570,39262,43200,47394,51850,56578,61584,66878,72466,78358,84560,91082,97930

%N Write the numbers from 1 to n^2 in a spiraling square; a(n) is the total of the sums of the two diagonals.

%C If n is odd, n^2 is counted twice.

%H Harry J. Smith, <a href="/A059924/b059924.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

%F a(n) = 3a(n-1)-2a(n-2)-2a(n-3)+3a(n-4)-a(n-5), a(0) = 0, a(1) = 2, a(2) = 10, a(3) = 34, a(4) = 80.

%F a(n) = (16*n^3 - 6*n^2 + 8*n + 3 - 3*(-1)^n)/12. - _Frank Ellermann_, Mar 16 2002

%F O.g.f.: (2*x+4*x^2+8*x^3+2*x^4)/(1-3*x+2*x^2+2*x^3-3*x^4+x^5)=(2*x+4*x^2+8*x^3+2*x^4)/((1-x)^4*(1+x)). - _Eric Werley_, Jun 30 2011

%e Write the numbers from 1 to 16 like this:

%e .

%e    1---2---3---4

%e                |

%e   12--13--14   5

%e    |       |   |

%e   11  16--15   6

%e    |           |

%e   10---9---8---7

%e .

%e The two diagonals add to 36 and 44, so a(4) = 36 + 44 = 80.

%t LinearRecurrence[{3,-2,-2,3,-1},{0,2,10,34,80},40] (* _Harvey P. Dale_, Mar 23 2012 *)

%o (PARI) a(n) = { (16*n^3 - 6*n^2 + 8*n + 3 - 3*(-1)^n)/12 } \\ _Harry J. Smith_, Jun 30 2009

%K easy,nice,nonn

%O 0,2

%A _Fabian Rothelius_, Feb 10 2001

%E Corrected and extended by _Eric Werley_, Jun 30 2011