login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Integers related to the volume k(n) of a unit hypersphere in n dimensions.
2

%I #48 Feb 07 2021 00:49:32

%S 0,2,2,4,4,8,6,16,8,32,10,64,12,128,14,256,16,512,18,1024,20,2048,22,

%T 4096,24,8192,26,16384,28,32768,30,65536,32,131072,34,262144,36,

%U 524288,38,1048576,40,2097152,42,4194304,44,8388608,46,16777216,48,33554432,50,67108864,52,134217728

%N Integers related to the volume k(n) of a unit hypersphere in n dimensions.

%C These numbers are arranged to give the simplest integers I could find with the n/2-symmetry the numbers show.

%C In his classic book on the n-dimensional geometry, Sommerville (1958, originally 1929) proved that the volume k(n) of a unit hypersphere in n-dimensions satisfies the recursion k(n) = k(n-1)*beta((n+1)/2, 1/2) for n >= 2 with k(1) = 2. Of course, k(n) = Pi^(n/2)/Gamma((n/2) + 1). The author of this sequence tries to find an integer sequence that is related to the volume sequence k(n) in some way. - _Petros Hadjicostas_, Feb 06 2021

%D D. M. Y. Sommerville, An Introduction of the Geometry of N dimensions, Dover Publications, 1958, pp. 135-137. [Corrected by _Petros Hadjicostas_, Feb 06 2021]

%H G. C. Greubel, <a href="/A138219/b138219.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Volume_of_an_n-ball">Volume of an n-ball</a>.

%F a(n) = n*k(n)/f(n), where k(0) = 1, k(1) = 2, k(n) = k(n-1)*Beta((n+1)/2, 1/2) for n >= 2, and f(n) = Pi^floor(n/2)/If[Mod[n, 2] == 0, (n/2)!, (n - 2)!!] with (-1)!! := 1. (For n >= 1, k(n) is the volume of an n-dimensional unit hypersphere; k(n) = Pi^(n/2)/Gamma(n/2 + 1).) [Modified by _Petros Hadjicostas_, Feb 06 2021]

%F From _Petros Hadjicostas_, Feb 06 2021: (Start)

%F a(n) = n if n is even, and = 2^((n+1)/2) if n is odd.

%F O.g.f.: 2*(1+x-2*x^2-2*x^3+x^4)/((1-x^2)^2*(1-2*x^2)). (End)

%e From _Petros Hadjicostas_, Feb 06 2021: (Start)

%e k(0) = 1, k(1) = 2, k(2) = Pi, k(3) = (4/3)*Pi^2, k(4) = Pi^2/2, k(5) = (8/15)*Pi^2, k(6) = Pi^3/6, k(7) = 16*Pi^3/105, k(8) = Pi^4/24, k(9) = 32*Pi^4/945, ...

%e f(0) = 1, f(1) = 1, f(2) = Pi, f(3) = Pi, f(4) = Pi^2/2, f(5) = Pi^2/3, f(6) = Pi^3/6, f(7) = Pi^3/15, f(8) = Pi^4/24, f(9) = Pi^4/105, ...

%e a(0) = 0*k(0)/f(0) = 0, a(1) = 1*k(1)/f(1) = 2, a(2) = 2*k(2)/f(2) = 2, a(3) = 3*k(3)/f(3) = 4, a(4) = 4*k(4)/f(4) = 4, a(5) = 5*k(5)/f(5) = 8, a(6) = 6*k(6)/f(6) = 6, a(7) = 7*k(7)/f(7) = 16, a(8) = 8*k(8)/f(8) = 8, a(9) = 9*k(9)/f(9) = 32, ... (End)

%t (* First Program *)

%t (* odd factorial function *)

%t b[n_]:= b[n] =If[n==0, 1, (2*n-1)*b[n-1]]; Table[b[n], {n, 0, 10}];

%t (* Pi factor function *)

%t f[n_]:= f[n] =Pi^Floor[n/2]/If[Mod[n, 2]==0, (n/2)!, b[Floor[n/2]]];

%t Table[f[n], {n, 0, 10}];

%t (* volume factors from Sommerville, pp. 136-137 *)

%t k[n_]:= k[n] =If[n<2, n+1, Beta[(n+1)/2, 1/2]*k[n-1]];

%t Table[k[n], {n, 0, 10}];

%t (* integer volume numbers *)

%t Table[n*k[n]/f[n], {n,0,60}]

%t (* Second Program *)

%t Table[((1+(-1)^n)*n + (1-(-1)^n)*Sqrt[2]^(n+1) )/2 , {n, 0, 55}] (* _G. C. Greubel_, Feb 06 2021 *)

%o (Sage) [((1+(-1)^n)*n + (1-(-1)^n)*sqrt(2)^(n+1) )/2 for n in (0..55)] # _G. C. Greubel_, Feb 06 2021

%Y Cf. A001147, A114348.

%K nonn

%O 0,2

%A _Roger L. Bagula_, May 05 2008

%E Various sections edited by _Petros Hadjicostas_, Feb 06 2021