A239608 - OEIS

%I #34 Nov 01 2025 19:50:34

%S 0,1,676,9801,59536,235225,715716,1825201,4096576,8346321,15760900,

%T 27994681,47279376,76545001,119552356,181037025,266864896,384199201,

%U 541679076,749609641,1020163600,1367594361,1808460676,2361862801,3049690176,3896880625,4931691076

%N a(n) = sin( arcsin(n)- 2*arccos(n) )^2.

%C The terms are integers.

%C This is assuming the "standard branch" of arcsin and arccos, where sin(arccos(n)) = cos(arcsin(n)) = sqrt(1-n^2). - _Robert Israel_, May 25 2014

%H Vincenzo Librandi, <a href="/A239608/b239608.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _Colin Barker_, May 24 2014: (Start)

%F a(n) = n^2*(3-4*n^2)^2.

%F G.f.: -x*(x+1)*(x^4+668*x^3+4422*x^2+668*x+1) / (x-1)^7. (End)

%F a(n) = A144129(n)^2. - _Robert Israel_, May 25 2014

%t G[n_, a_, b_] := G[n, a, b] = Sin[a ArcSin[ n] + b ArcCos[n]]^2 // ComplexExpand // FullSimplify; Table[G[n, 1, -2], {n, 0, 43}]

%t CoefficientList[Series[- x (x + 1) (x^4 + 668 x^3 + 4422 x^2 + 668 x + 1)/(x - 1)^7, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 30 2014 *)

%t Table[n^2*(3-4*n^2)^2,{n,0,30}] (* _Harvey P. Dale_, Aug 05 2016 *)

%o (PARI) vector(100, n, round(sin(asin(n-1) - 2*acos(n-1))^2)) \\ _Colin Barker_, May 24 2014

%o (Magma) [n^2*(3-4*n^2)^2 : n in [0..50]]; // _Vincenzo Librandi_, May 30 2014

%Y Cf. A239607, A239609, A239610.

%K nonn,easy

%O 0,3

%A _José María Grau Ribas_, Mar 22 2014