A394090 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].

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, 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, 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

Offset

1,1

Comments

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.

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.

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

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

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

It is conjectured that this constant is transcendental.

Links

Table of n, a(n) for n=1..100.

Example

2.277552113851191588974737802127065126412089221...

Mathematica

$MaxExtraPrecision = 100;
I[a_?NumericQ] := N[Integrate[(Sin[Pi*x/2]^(1/a) - Sqrt[2x - x^2])^2, {x, 0, 1}, WorkingPrecision -> 50]];
res = FindMinimum[I[a], {a, 2}, WorkingPrecision -> 50];
optimalA = a /. res[[2]];
RealDigits[N[optimalA, 80]]

Prog

(PARI) f(a) = intnum(x=0, 1, (sin((Pi/2)*x)^(1/a) - sqrt(2*x-x^2))^2);
solve(a=2.27, 2.28, f'(a)) \\ Hugo Pfoertner, Mar 16 2026

Crossrefs

Cf. A000796.

Sequence in context: A347073 A261710 A064288 * A054085 A357412 A021443

Adjacent sequences: A394087 A394088 A394089 * A394091 A394092 A394093

Keyword

nonn,cons

Author

Minyan Liu, Mar 10 2026

Status

approved