A394090 - OEIS

%I #18 May 26 2026 00:47:43

%S 2,2,7,7,5,5,2,1,1,3,8,5,1,1,9,1,5,8,8,9,7,4,7,3,7,8,0,2,1,2,7,0,6,5,

%T 1,2,6,4,1,2,0,8,9,2,2,1,0,9,0,4,6,6,8,6,8,6,3,7,8,2,8,5,4,2,4,7,2,9,

%U 2,4,5,5,9,2,8,0,7,4,6,8,2,6,9,0,2,9,7,3,3,8,7,5,9,6,7,0,0,7,3,1

%N Decimal expansion of the parameter c that minimizes the integral of squared distance between sin((Pi/2)x)^(1/c) and the quarter-circle sqrt(2x-x^2) on [0,1].

%C Decimal expansion of the unique positive real number c that minimizes the integral I(c) = integral_{x=0..1} (sin((Pi/2)x)^(1/c) - sqrt(2x-x^2))^2 dx.

%C Geometrically, the curve y = sin((Pi/2)x)^(1/c) is a one-parameter family that interpolates between a quarter-circle (at this optimal c) and a unit square as c -> infinity.

%C The function sqrt(2x-x^2) is the lower-right quarter of the unit circle centered at (1,0).

%C The optimal c gives the best least-squares fit to the quarter-circle in this family.

%C The value c is the unique solution to d/dx I(x) = 0, which must be solved numerically.

%C It is conjectured that this constant is transcendental.

%e 2.277552113851191588974737802127065126412089221...

%t $MaxExtraPrecision = 100;

%t I[a_?NumericQ] := N[Integrate[(Sin[Pi*x/2]^(1/a) - Sqrt[2x - x^2])^2, {x, 0, 1}, WorkingPrecision -> 50]];

%t res = FindMinimum[I[a], {a, 2}, WorkingPrecision -> 50];

%t optimalA = a /. res[[2]];

%t RealDigits[N[optimalA, 80]]

%o (PARI) f(a) = intnum(x=0, 1, (sin((Pi/2)*x)^(1/a) - sqrt(2*x-x^2))^2);

%o solve(a=2.27, 2.28, f'(a)) \\ _Hugo Pfoertner_, Mar 16 2026

%Y Cf. A000796.

%K nonn,cons

%O 1,1

%A _Minyan Liu_, Mar 10 2026