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A138219
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Integers related to the volume k(n) of a unit hypersphere in n dimensions.
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2
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0, 2, 2, 4, 4, 8, 6, 16, 8, 32, 10, 64, 12, 128, 14, 256, 16, 512, 18, 1024, 20, 2048, 22, 4096, 24, 8192, 26, 16384, 28, 32768, 30, 65536, 32, 131072, 34, 262144, 36, 524288, 38, 1048576, 40, 2097152, 42, 4194304, 44, 8388608, 46, 16777216, 48, 33554432, 50, 67108864, 52, 134217728
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OFFSET
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0,2
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COMMENTS
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These numbers are arranged to give the simplest integers I could find with the n/2-symmetry the numbers show.
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
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REFERENCES
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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]
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LINKS
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FORMULA
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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]
a(n) = n if n is even, and = 2^((n+1)/2) if n is odd.
O.g.f.: 2*(1+x-2*x^2-2*x^3+x^4)/((1-x^2)^2*(1-2*x^2)). (End)
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EXAMPLE
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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, ...
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, ...
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)
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MATHEMATICA
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(* First Program *)
(* odd factorial function *)
b[n_]:= b[n] =If[n==0, 1, (2*n-1)*b[n-1]]; Table[b[n], {n, 0, 10}];
(* Pi factor function *)
f[n_]:= f[n] =Pi^Floor[n/2]/If[Mod[n, 2]==0, (n/2)!, b[Floor[n/2]]];
Table[f[n], {n, 0, 10}];
(* volume factors from Sommerville, pp. 136-137 *)
k[n_]:= k[n] =If[n<2, n+1, Beta[(n+1)/2, 1/2]*k[n-1]];
Table[k[n], {n, 0, 10}];
(* integer volume numbers *)
Table[n*k[n]/f[n], {n, 0, 60}]
(* Second Program *)
Table[((1+(-1)^n)*n + (1-(-1)^n)*Sqrt[2]^(n+1) )/2 , {n, 0, 55}] (* G. C. Greubel, Feb 06 2021 *)
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PROG
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(Sage) [((1+(-1)^n)*n + (1-(-1)^n)*sqrt(2)^(n+1) )/2 for n in (0..55)] # G. C. Greubel, Feb 06 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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