A072345 -id:A072345 - cp's OEIS frontend

A074454 Consider volume 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 volume.

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

Offset: 1

Robert G. Wilson v, Aug 22 2002

The dimension is given in A074455.

If you set v(n) = Pi^(n/2)/(n/2)! and s(n) = n*Pi^(n/2)/(n/2)! and then plot {6.283*v(n-2), s(n)} for 0<=n<=20, the two curves are almost identical.

			5.277768021113400997282145864172846387529999284510173567761637340214864\
12730547017110062048407258401284645...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

Cf. A072345, A072346.

d = x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105]; First[ RealDigits[ Pi^(d/2)/(d/2)!]][[1 ;; 99]] (* Jean-François Alcover, Apr 12 2013 *)

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

A057649 Nearest integer to ratio of volume of n-dimensional cube of side 2 to volume of n-dimensional ball of radius 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 27, 63, 155, 402, 1087, 3068, 8996, 27340, 85905, 278485, 929713, 3191201, 11245603, 40631628, 150342633, 569071745, 2201355134, 8694775795, 35035655103, 143916920872, 602218550964, 2565370007349, 11118142868084, 48994958103507

Offset: 0

N. J. A. Sloane, Sep 06 2003

nonn

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

Cf. A072168, A072345, A072346.

a(n) = 2^n*GAMMA(n/2+1)/Pi^(n/2).

A072168 Floor of ratio of volume of n-dimensional cube of side 2 to volume of n-dimensional ball of radius 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 12, 27, 63, 155, 401, 1086, 3067, 8995, 27340, 85905, 278484, 929713, 3191200, 11245602, 40631627, 150342632, 569071745, 2201355133, 8694775794, 35035655103, 143916920871, 602218550964, 2565370007348, 11118142868084, 48994958103507

Offset: 0

N. J. A. Sloane, Sep 06 2003

nonn

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

Cf. A057649, A072345, A072346.

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

A375966 Powers of 3 alternating with powers of 4.

Original entry on oeis.org

1, 1, 3, 4, 9, 16, 27, 64, 81, 256, 243, 1024, 729, 4096, 2187, 16384, 6561, 65536, 19683, 262144, 59049, 1048576, 177147, 4194304, 531441, 16777216, 1594323, 67108864, 4782969, 268435456, 14348907, 1073741824, 43046721, 4294967296, 129140163, 17179869184

Offset: 0

Paul Curtz, Sep 04 2024

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-12).

Cf. A000244 and A000302 interleaved.

Cf. A000035, A059841, A072345, A098293.

seq[len_] := Module[{m = Ceiling[len/2] - 1}, Riffle @@ Map[#^Range[0, m] &, {3, 4}]]; seq[36] (* Amiram Eldar, Sep 05 2024 *)

def A375966(n): return 1<<(n^1) if n&1 else 3**(n>>1) # Chai Wah Wu, Sep 24 2024

a(n) = 7*a(n-2) - 12*a(n-4) for n >= 4.
From Stefano Spezia, Sep 06 2024: (Start)
G.f.: (1 + x - 4*x^2 - 3*x^3)/((1 - 2*x)*(1 + 2*x)*(1 - 3*x^2)).
a(n) = (4*3^(n/2)*A059841(n) - (-2)^n + 2^n)/4.
E.g.f.: cosh(sqrt(3)*x) + cosh(x)*sinh(x). (End)

A377825 Number of distinct permutations of the terms of the n-th row of Pascal's triangle with alternating signs.

Original entry on oeis.org

1, 2, 3, 24, 30, 720, 630, 40320, 22680, 3628800, 1247400, 479001600, 97297200, 87178291200, 10216206000, 20922789888000, 1389404016000, 6402373705728000, 237588086736000, 2432902008176640000, 49893498214560000, 1124000727777607680000, 12623055048283680000

Offset: 0

Ryan Jean, Nov 08 2024

Note that for any given n, there are n+1 terms in that row.

			For n = 0, a(0) = 1 since there is just one term.
For n = 1, the signed row terms are {1, -1} so a(1) = 2 permutations.
For n = 2, the signed row terms are {1, -2, 1} which have only a(2) = 3 distinct permutations.
For n = 3, the signed row terms are {1, -3, 3, -1} which have a(3) = 24 permutations.

Paolo Xausa, Table of n, a(n) for n = 0..400

Bisections are: A007019, A010050.

Cf. A007318, A072345, A142150.

seq((n+1)! / (2^((n*(1+(-1)^n)) / 4)), n=0..22); # Georg Fischer, Dec 19 2024

A377825[n_] := (n+1)!/2^((n*(1 + (-1)^n))/4); Array[A377825, 25, 0] (* Paolo Xausa, Dec 20 2024 *)

a(n) = (n+1)! / (2^((n*(1+(-1)^n)) / 4)).
E.g.f.: 2*(x^6+x^5-4*x^3-3*x^2+4*x+2)/((x-1)^2*(x+1)^2*(x^2-2)^2). - Alois P. Heinz, Nov 09 2024
a(n) = (n+1)!/A072345(n-1) for n > 0. - Stefano Spezia, Nov 09 2024
Sum_{n>=0} 1/a(n) = cosh(1) + sinh(sqrt(2))/sqrt(2) - 1. - Amiram Eldar, Dec 25 2024

a(22) corrected by Georg Fischer, Dec 19 2024

cp's OEIS Frontend

A074454 Consider volume 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 volume.

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

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

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Maple

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

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A057649 Nearest integer to ratio of volume of n-dimensional cube of side 2 to volume of n-dimensional ball of radius 1.

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Author

Keywords

References

Crossrefs

Formula

A072168 Floor of ratio of volume of n-dimensional cube of side 2 to volume of n-dimensional ball of radius 1.

Views

Author

Keywords

References

Crossrefs

Formula

A375966 Powers of 3 alternating with powers of 4.

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Programs

Mathematica

Python

Formula

A377825 Number of distinct permutations of the terms of the n-th row of Pascal's triangle with alternating signs.