A185950 - OEIS

%I #35 Sep 08 2022 08:45:55

%S -1,2,13,32,59,94,137,188,247,314,389,472,563,662,769,884,1007,1138,

%T 1277,1424,1579,1742,1913,2092,2279,2474,2677,2888,3107,3334,3569,

%U 3812,4063,4322,4589,4864,5147,5438,5737,6044,6359,6682,7013,7352,7699,8054,8417,8788,9167,9554,9949,10352,10763,11182,11609

%N a(n) = 4*n^2 - n - 1.

%C Write the sequence A023443 in a clockwise spiral. a(n) is on the y-axis.

%C a(n) mod 9 = period 9: repeat [8,2,4,5,5,4,2,8,4] = A182868(n+2) mod 9.

%H G. C. Greubel, <a href="/A185950/b185950.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A176126(4*n-1) = A054556(n+1) - 2 = A033991(n) - 1.

%F a(n) = a(n-1) + 8*n - 5.

%F a(n) = 2*a(n-1) - a(n-2) + 8.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: ( 1-5*x-4*x^2 ) / (x-1)^3. - _R. J. Mathar_, Feb 10 2011

%F E.g.f.: (4*x^2 + 3*x - 1)*exp(x). - _G. C. Greubel_, Jul 23 2017

%e   11--12--13--14--15

%e    |               |

%e   10   1---2---3  16

%e    |   |       |   |

%e    9   0-(-1)  4  17

%e    |           |   |

%e    8---7---6---5  18

%p A185950:=n->4*n^2-n-1: seq(A185950(n), n=0..100); # _Wesley Ivan Hurt_, Jan 30 2017

%t Table[4n^2-n-1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{-1,2,13},60] (* _Harvey P. Dale_, May 22 2015 *)

%o (Magma)[-1-n+4*n^2: n in [0..80]]; // _Vincenzo Librandi_, Feb 08 2011

%o (PARI) a(n)=4*n^2-n-1 \\ _Charles R Greathouse IV_, Dec 21 2011

%o (Haskell)

%o a185950 n = (4 * n - 1) * n - 1 -- _Reinhard Zumkeller_, Aug 14 2013

%Y Cf. A033951, A182868.

%K sign,easy

%O 0,2

%A _Paul Curtz_, Feb 07 2011