A193506 Decimal expansion of bean curve perimeter.

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

Offset: 1

Author

Jean-François Alcover, Jul 29 2011

Keywords

• nonn
• cons

Examples

			3.750213645...
		

Links

• Eric Weisstein's World of Mathematics, Bean Curve

Crossrefs

Cf. A193505 (area).

Cf. A336501 (decimal expansion of the lima bean curve).

Programs

Mathematica

f[x_, y_] = x^4 + x^2*y^2 + y^4 - x*(x^2 + y^2); x1 = 1/3; x2 = 5/6; sx = Solve[f[x, y] == 0, x]; sy = Solve[f[x, y] == 0, y]; g1[y_] = x /. sx[[3]]; g2[y_] = x /. sx[[4]]; f[x_] = y /. sy[[4]]; p1 = NIntegrate[ Sqrt[1 + g1'[y]^2], {y, 0, f[x1]}, WorkingPrecision -> 120]; p2 = NIntegrate[ Sqrt[1 + f'[x]^2], {x, x1, x2}, WorkingPrecision -> 120]; p3 = NIntegrate[ Sqrt[1 + g2'[y]^2], {y, 0, f[x2]}, WorkingPrecision -> 120]; Take[ RealDigits[2*(p1+p2+p3)][[1]], 105]
Take[RealDigits[9/2 + NIntegrate[2 Sqrt[1 + (1 - 2 x + (1 + 3 x - 6 x^2)/Sqrt[1 + (2 - 3 x) x])^2/(8 x (1 - x + Sqrt[1 + (2 - 3 x) x]))] - 1/Sqrt[x] - 1/(2 (1 - x)^(3/4)) - 3/(8 (1 - x)^(1/4)), {x, 0, 1}, WorkingPrecision -> 220, PrecisionGoal -> 110, MaxRecursion -> 50]][[1]], 110] (* Eric W. Weisstein, Jul 23 2020 *)