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A053381
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Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.
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6
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1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3
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OFFSET
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0,2
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COMMENTS
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The corresponding terms for a 2n-sphere are all 0 ("you can't comb the hair on a billiard ball"). The "3" and "7" come from the quaternions and octonions.
b(n) = a(n-1): b(2^e) = ((e+1) idiv 4) + 2^((e+1) mod 4) - 1, b(p^e) = 1, p>2. - Christian G. Bower May 18, 2005.
a(n-1) is multiplicative. - Christian G. Bower, Jun 03 2005
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REFERENCES
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J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..10000
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FORMULA
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Let f(n) be the number of linearly independent smooth nowhere zero vector fields on an n-sphere. Then f(n) = 2^c + 8d - 1 where n+1 = (2a+1) 2^b and b = c+4d and 0 <= c <= 3. f(n) = 0 if n is even.
a((2*n+1)*2^p-1) = A047530(p+1), p >= 0 and n >= 0. a(2*n) = 1, n >= 0, and a(2^p-1) = A047530(p+1), p >= 0. [Johannes W. Meijer, Jun 7 2011]
a(n) = A209675(n+1) - 1. [Reinhard Zumkeller, Mar 11 2012]
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MAPLE
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with(numtheory): for n from 1 to 601 by 2 do c := irem(ifactors(n+1)[2, 1, 2], 4): d := iquo(ifactors(n+1)[2, 1, 2], 4): printf(`%d, `, 2^c+8*d-1) od:
nmax:=101: A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do A053381((2*n+1)*2^p-1) := A047530(p+1): od: od: seq(A053381(n), n=0..nmax); # [Johannes W. Meijer, Jun 7 2011, Revised Jan 29 2013]
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MATHEMATICA
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a[n_] := Module[{b, c, d, rho, n0}, n0 = 2*n; b = 0; While[BitAnd[n0, 1] == 0, n0 /= 2; b++]; c = BitAnd[b, 3]; d = (b - c)/4; rho = 2^c + 8*d; Return[rho - 1]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, May 16 2013, translated from c *)
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PROG
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(C:) int MaxLinInd(int n){ /* Returns max # linearly indep smooth nowhere zero * vector fields on S^{n-1}, n=1, 2, ... */ int b, c, d, rho; b = 0; while((n & 1)==0){ n /= 2; b++; } c = b & 3; d = (b - c)/4; rho = (1 << c) + 8*d; return( rho - 1); }
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CROSSREFS
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For another version see A003484. Cf. A047680, A001676.
Cf. A047530, A220466.
Sequence in context: A136011 A021991 A112132 * A038712 A065745 A117677
Adjacent sequences: A053378 A053379 A053380 * A053382 A053383 A053384
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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wds(AT)research.nj.nec.com (Warren Smith), Jan 06 2000
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EXTENSIONS
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More terms from James A. Sellers, Jun 01 2000
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STATUS
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approved
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