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

1,1

Links

Table of n, a(n) for n=1..110.

Eric Weisstein's World of Mathematics, Bean Curve

Example

3.750213645...

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 *)

Crossrefs

Cf. A193505 (area).

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

Sequence in context: A331733 A301755 A302558 * A086242 A322931 A096627

Adjacent sequences: A193503 A193504 A193505 * A193507 A193508 A193509

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 29 2011

Status

approved