A325629 Floor of number of n-dimensional degrees in an n-sphere.

2, 360, 41252, 3712766, 283634468, 19145326633, 1170076174384, 65816784809141, 3447793362911604, 16969079580805447, 7901760333122072321, 350023289756266797348, 14816864219294689084225

Offset: 0

Author

Eliora Ben-Gurion, Sep 07 2019

Keywords

• nonn

Comments

Only the 0th and 1st terms of this sequence are exact values of n-degrees in an n-sphere, by definition. The 0-sphere, being 2 disconnected points at the ends of a segment, is trivial.

The number of degrees, minutes, seconds in an n-sphere is designed to approximate the size of an n-cube, m^n units in size, as m becomes increasingly small, observed from the center of the sphere. This makes a degree Pi/180 of a radian, a square degree (Pi/180)^2 of a steradian, a cubic degree (Pi/180)^3 of a 3-radian, etc.

The sequence has a maximum value at n = 20626 with a value of 1.3610489172...*10^4479, too large to be written here. I conjecture that the peak value of the function analytically is somewhere near 64800/Pi = 20626.48062...

At n = 56058 the sequence has a value of 281 (actual number 281.4089), meaning the 56058-dimensional sphere has less than 360 degrees. At n = 56070, the function has a value of 0.6978855, turning the rest of the sequence into a string of zeros.

An "N-sphere" is located in an N+1-dimensional space, 1-sphere being a circle, 2-sphere being an ordinary sphere, and so on.

From Jon E. Schoenfield, Sep 07 2019: (Start)

The maximum value of the continuous function is 1.361052727810610001492173640278424460497...*10^4479 and it occurs at 20626.48061662940750570152124725484602696... which is close to 64800/Pi, but it's actually 64799.99997461521504462375443773494034381.../Pi. That numerator appears to be 64800 - z/64800 + (27/10) * z^2 / 64800^3 - ... where z = zeta(2) = Pi^2 / 6. (End)

Examples

			Number of cubic degrees in a 3-sphere:
Surface area of a 3-sphere: 2*Pi^((3+1)/2) / ((3+1)/2 - 1)! = 2*Pi^2 / (2-1)! = 2*Pi^2.
Cubic degrees: 2*Pi^2 * (180/Pi)^3 = 11664000 / Pi = 3712766.512...
Number of tesseractic degrees in a 4-sphere:
Surface area of a 4-sphere: 2*Pi^((4+1)/2) / Gamma(5/2) = 2*Pi^(5/2) / (3*Pi^(1/2)/4) = 8*Pi^2/3.
Tesseractic degrees: 8*Pi^2/3 * (180/Pi)^4 = 2799360000 / Pi^2 = 283634468.641...
		

Links

• Eric Weisstein's World of Mathematics, Solid Angle.
• Wikipedia, N-sphere
• Wikipedia, Particular values of the gamma function

Crossrefs

Surface area of k-dimensional sphere for k=2..8: A019692, A019694, A164102, A164104, A091925, A164107, A164109.

Cf. A125560.

Formula

a(n) = floor((2*Pi^((n+1)/2)/((n+1)/2-1)!)/(Pi/180)^n).

a(n) = floor((2*Pi^((n+1)/2)/(Gamma((n+1)/2)))/(Pi/180)^n).

a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/((n+1)/2-1)!).

a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/(Gamma((n+1)/2))).