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Search: id:A138219
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| A138219 |
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Integer volume numbers for n dimensions from Sommervillie recursion formula: k(n)=k(n-1)*Beta[(n+1)/2,1/2]. |
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+0
1
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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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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These numbers are arranged to give the simplest integers I could find
with the n/2 symmetry the numbers show.
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REFERENCES
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D. M. Y. Sommerville, The Elements of Non-Euclidean Geometry, Dover Publications, 1958, pp. 135-137. MR0100246 (20 #6679)
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FORMULA
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k(0)=1;k(1)=2; k(n)=k(n-1)*Beta((n+1)/2,1/2); f(n)=Pi^Floor[n/2]/If[Mod[n, 2] == 0, (n/2)!, odd_factorial[Floor[n/2]]] out[n]=n*k(n)/f(n).
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MATHEMATICA
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Clear[a, f, k] (* odd factorial function*) a[0] = 1; a[n_] := a[n] = (2*n - 1)*a[n - 1]; Table[a[n], {n, 0, 10}]; (* Pi factor function*) f[n_] := f[n] = Pi^Floor[n/2]/If[Mod[n, 2] == 0, (n/2)!, a[Floor[n/2]]] Table[f[n], {n, 0, 10}]; (*volume factors from Sommerville, page 136 - 137*) k[0] = 1; k[1] = 2; k[n_] := k[n] = k[n - 1]*Beta[(n + 1)/2, 1/2] Table[k[n], {n, 0, 10}]; (* integer volume numbers*) Table[n*k[n]/f[n], {n, 0, 30}]
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CROSSREFS
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Cf. A114446, A114348.
Sequence in context: A115209 A139210 A008330 this_sequence A100835 A120541 A059867
Adjacent sequences: A138216 A138217 A138218 this_sequence A138220 A138221 A138222
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 05 2008
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