A072479 -id:A072479 - cp's OEIS frontend

A072345 Volume of n-dimensional sphere of radius r is V_nr^n = Pi^(n/2)r^n/(n/2)! = C_nPi^floor(n/2)r^n; sequence gives numerator of C_n.

1, 2, 1, 4, 1, 8, 1, 16, 1, 32, 1, 64, 1, 128, 1, 256, 1, 512, 1, 1024, 1, 2048, 1, 4096, 1, 8192, 1, 16384, 1, 32768, 1, 65536, 1, 131072, 1, 262144, 1, 524288, 1, 1048576, 1, 2097152, 1, 4194304, 1, 8388608, 1, 16777216, 1, 33554432, 1, 67108864, 1, 134217728, 1, 268435456

Offset: 0

N. J. A. Sloane, Jul 31 2002

Answer to question of how to extend the sequence 1, 2 r, Pi r^2, 4 Pi r^3 / 3, Pi^2 r^4 / 2, ...

Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1). - see A072478/A072479.

			Sequence of C_n's begins 1, 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, 1/24, 32/945, 1/120, 64/10395, ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.

N. Cakic, D. Letic, and B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010).
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Ball
Eric Weisstein's World of Mathematics, Four-Dimensional Geometry
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A072346, A072478, A072479.

[2^((n+1) div 2)*(1-(-1)^n)/2+(1+(-1)^n)/2 : n in [0..100]]; // Wesley Ivan Hurt, Jan 10 2017

seq(seq(k^n, k=1..2), n=1..28); # Zerinvary Lajos, Jun 29 2007

f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 55} ]
Riffle[2^Range[30],1,{1,-1,2}] (* or *) LinearRecurrence[{0,3,0,-2},{1,2,1,4},60] (* Harvey P. Dale, Oct 16 2013 *)
CoefficientList[ Series[(-2x^3 - 2x^2 + 2x + 1)/(2x^4 - 3x^2 + 1), {x, 0, 56}], x] (* Robert G. Wilson v, Jul 31 2018 *)

def A072345(n): return 2<<(n>>1) if n&1 else 1 # Chai Wah Wu, Sep 24 2024

1 if n even, 2^((n+1)/2) if n odd.
a(n) = 3*a(n-2)-2*a(n-4). G.f.: (1+2*x-2*x^2-2*x^3)/((1-x)*(1+x)*(1-2*x^2)). [Colin Barker, Sep 04 2012]
a(n) = 2^((n+1)/2)*(1-(-1)^n)/2+(1+(-1)^n)/2. - Wesley Ivan Hurt, Jan 10 2017
E.g.f.: sqrt(2)*sinh(sqrt(2)*x) + cosh(x). - Ilya Gutkovskiy, Mar 16 2017

A072478 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives numerator of S_n.

Original entry on oeis.org

0, 2, 2, 4, 2, 8, 1, 16, 1, 32, 1, 64, 1, 128, 1, 256, 1, 512, 1, 1024, 1, 2048, 1, 4096, 1, 8192, 1, 16384, 1, 32768, 1, 65536, 1, 131072, 1, 262144, 1, 524288, 1, 1048576, 1, 2097152, 1, 4194304, 1, 8388608, 1, 16777216, 1, 33554432, 1, 67108864, 1

Offset: 0

N. J. A. Sloane, Aug 02 2002

Answer to question of how to extend the sequence 0, 2, 2 Pi r, 4 Pi r^2, 2 Pi^2 r^3, ...
Volume of n-dimensional sphere of radius r is V_n*r^n - see A072345/A072346.
a(2*n-1) = 2^n and for n>2 a(2*n)=1.
Denominator of the rational coefficient of integral_{x>0} exp(-x^2)*x^n. - Jean-François Alcover, Apr 23 2013
From Ilya Gutkovskiy, Aug 02 2016: (Start)
Numerator of n/Gamma(n/2+1).
More generally, the ordinary generating function for the surface area of the n-dimensional sphere of radius r is 2*x*(1 + exp(Pi*r^2*x^2)*Pi*r*x + exp(Pi*r^2*x^2)*Pi*r*erf(sqrt(Pi)*r*x)*x) =  2*x + 2*Pi*r*x^2 + 4*Pi*r^2*x^3 + 2*Pi^2*r^3*x^4 + (8*Pi^2*r^4/3)*x^5 + Pi^3*r^5*x^6 + ... (End)

			Sequence of S_n's begins 0, 2, 2, 4, 2, 8/3, 1, 16/15, 1/3, 32/105, 1/12, 64/945, ...

N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.

Colin Barker, Table of n, a(n) for n = 0..1000
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Ball
Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Four-Dimensional Geometry
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A072479. A072478(n)/A072479(n) = n*A072345(n)/A072346(n).

f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 52}]
CoefficientList[Series[x (2 + 2 x - 2 x^2 - 4 x^3 - x^5 + 2 x^7)/(1 - 3 x^2 + 2 x^4), {x, 0, 52}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{0,3,0,-2},{0,2,2,4,2,8,1,16,1},60] (* Harvey P. Dale, May 30 2018 *)

concat(0, Vec(x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7)/(1-3*x^2+2*x^4) + O(x^100))) \\ Colin Barker, Aug 01 2016

From Colin Barker, Sep 04 2012: (Start)
a(n) = 3*a(n-2)-2*a(n-4) for n>4.
G.f.: x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7) / (1-3*x^2+2*x^4).
(End)
From Colin Barker, Aug 01 2016: (Start)
a(n) = (1+(-1)^n-2^((1+n)/2)*(-1+(-1)^n))/2 for n>4.
a(n) = 1 for n>4 and even.
a(n) = 2^((n+1)/2) for n>4 and odd.
(End)

More terms from Robert G. Wilson v, Aug 18 2002

A074457 Consider surface area of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 22 2002

			7.256946404860576780132838388690769236619017237183214857509879678777...

Nenad Cakic, Dusko Letic, and Branko Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Dusko Letic, Nenad Cakic, Branko Davidovic, and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Hypersphere.

Surface area is A074456. Cf. A072478, A072479, A074455.

RealDigits[ FindMinimum[ -n*Pi^(n/2)/(n/2)!, {n, 7}, WorkingPrecision -> 125] [[2, 1, 2]]] [[1]]
x /. FindRoot[ PolyGamma[x/2] == Log[Pi], {x, 7}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

Equals 2 + A074455.

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A074456 Consider surface area of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the resulting surface area.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Aug 22 2002

If you set v[n_] := Pi^(n/2)/(n/2)! and s[n_] := n*Pi^(n/2)/(n/2)! and then Plot[{6.283v[n - 2], s[n]}, {n, 0, 20}], the two curves are almost identical.

			33.1611944849620026918630240155829735800472328410872...

The dimension is given in A074455.

Cf. A072478, A072479, A074454.

area[d_] := d * Pi^(d/2)/Gamma[d/2 + 1]; area[x /. FindRoot[PolyGamma[x/2] == Log[Pi], {x, 7}, WorkingPrecision -> 120]] (* Amiram Eldar, Jun 08 2023 *)

Checked by Martin Fuller, Jul 12 2007

A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 259, 408, 512, 536, 481, 378, 264, 166, 94, 49, 24, 10, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the ceiling is A164083.
The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.
There are only 17 nonzero terms. - G. C. Greubel, Sep 10 2017

			Table of approximate real values before taking integer part.
========================
n (2*Pi)^n / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd ed., New York: Springer-Verlag, p. 9, 1993.
H. S. M. Coxeter, Regular Polytopes, 3rd ed., New York: Dover, 1973.
D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere.

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255.

A164081 := proc(n) (2*Pi)^n/(n-1)! ; floor(%) ; end: seq(A164081(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Floor[(2*Pi)^n/(n - 1)!], {n, 1, 100}] (* G. C. Greubel, Sep 10 2017 *)

for(n=1,100, print1(floor((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 10 2017

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

Definition corrected by R. J. Mathar, Sep 09 2009

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 260, 408, 513, 537, 482, 379, 264, 166, 95, 50, 24, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The floor of this real sequence is A164081, the ceiling is A164083.
The surface area of the n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/ A072479.
There are 18 nonzero terms in this sequence. - G. C. Greubel, Sep 11 2017

			Table of approximate real values before rounding up or down:
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.

A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* G. C. Greubel, Sep 11 2017 *)

for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 11 2017

a(n) = round(((2*Pi)^n)/(n-1)!).

Definition corrected by R. J. Mathar, Sep 09 2009

A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

7, 40, 125, 260, 409, 513, 537, 482, 379, 265, 167, 95, 50, 25, 11, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the floor is A164081.

The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.

			Table of approximate real values before rounding up.
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164082.

Table[Ceiling[(2Pi)^n/(n-1)!],{n,60}] (* Harvey P. Dale, Jul 30 2020 *)

a(n) = ceiling(((2*pi)^n)/(n-1)!).

Definition corrected - R. J. Mathar, Sep 09 2009

cp's OEIS Frontend

A072345 Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives numerator of C_n.

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A072478 Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives numerator of S_n.

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A074457 Consider surface area of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

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Mathematica

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A074456 Consider surface area of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the resulting surface area.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

A072345 Volume of n-dimensional sphere of radius r is V_nr^n = Pi^(n/2)r^n/(n/2)! = C_nPi^floor(n/2)r^n; sequence gives numerator of C_n.

A072478 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives numerator of S_n.