%I #24 Sep 08 2022 08:45:55

%S -1,4,17,38,67,104,149,202,263,332,409,494,587,688,797,914,1039,1172,

%T 1313,1462,1619,1784,1957,2138,2327,2524,2729,2942,3163,3392,3629,

%U 3874,4127,4388,4657,4934,5219,5512,5813,6122,6439,6764,7097,7438,7787,8144,8509,8882,9263,9652

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

%C First quadrisection of A176126(n). Take clockwise (square) spiral from A023443(n)=n-1: a(n) is on the negative x-axis. Fourth quadrisection (-1-n+4*n^2) is on the negative y-axis.

%C Conjecture: the 4 quadrisections of (the family) A064038, A160050, A176126, A178242 (see A181407) come from square spiral.

%C a(n) mod 9 has period 9: 8,4,8,2,4,5,5,4,2. a(n) mod 10 has period 10: 9,4,7,8,7,4,9,2,3,2. Each polynomial modulo some constant c has a period of length c (and perhaps shorter ones). - _Paul Curtz_ and _Bruno Berselli_, Feb 05 2011

%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).

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

%F a(n) = a(n-1) - 3 + 8*n.

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

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

%F G.f.: -(1 - 7*x - 2*x^2)/(1-x)^3. - _Bruno Berselli_, Feb 05 2011

%t f[n_]:=-1+n+4*n^2;f[Range[0,100]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2011 *)

%o (Magma) [-1+n+4*n^2: n in [0..700] ] // _Vincenzo Librandi_, Feb 01 2011

%o (PARI) a(n)=-1+n+4*n^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%K sign,easy

%O 0,2

%A _Paul Curtz_, Feb 01 2011