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A231190 Numerator of abs(n-8)/(2*n), n >= 1. 4
7, 3, 5, 1, 3, 1, 1, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 4, 65, 33, 67, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Because 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*a(n)/b(n)) with gcd(a(n),b(n)) = 1, one has

  2*sin(Pi*4/n) = R(a(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/b(n)) =: rho(b(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.

  b(n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(b(n))) of degree delta(b(n)), with delta(k) = A055034(k). This degree delta(b(n)) is given in A231193(n), and if gcd(n,2) = 1 it coincides with the one for sin(2*Pi/n) given by A093819(n). See Theorem 3.9 of the I. Niven reference, pp. 37-38, which uses gcd(k, n) = 1. See also the Jan 09 2011 comment on A093819.

a(n) and b(n) = A232625(n) are the k=2 members of a family of pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1, relevant to determine the algebraic degree of 2*sin(Pi*2*k/n) from the trigonometric identity (used in the D. H. Lehmer and I. Niven references) 2*sin(Pi*2*k/n) = 2*cos(Pi*abs(n-4*k)/(2*n)) = 2*cos(Pi*p(k,n)/q(k,n)). This is R(p(k,n), x) (mod C(q(k,n), x)), with x = 2*cos(Pi/q(k,n)) =: rho(q(k,n)). The polynomials R and C have been used above. C(q(k,n), x) is the minimal polynomial of rho(q(k,n)) with degree delta(q(k,n)), which is then the degree, call it deg(k,n), of the integer 2*sin(Pi*2*k/n) in the number field Q(rho(q(k,n))). From Theorem 3.9 of the I. Niven reference deg(k,n) is, for given k, for those n with gcd(k, n) = 1 determined by A093819(n). In general deg(k,n) = A093819(n/gcd(k,n)). For the k=1 instance p(1,n) and q(1,n) see comments on A106609 and A225975.

REFERENCES

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.

FORMULA

a(n) = numerator(abs(n-8)/(2*n)), n >= 1.

a(n) = abs(n-8)/gcd(n-8, 16).

a(n) = abs(n-8) if n is odd; if n is even then a(n) = abs(n-8)/2 if n/2 == 1, 3, 5, 7 (mod 8), a(n) = abs(n-8)/4  if n/2 == 2, 6 (mod 8), a(n) = abs(n-8)/8 if n/2 == 0 (mod 8) and a(n) = abs(n-8)/16 if n == 4 (mod 8).

O.g.f.: 1+ x*(7 + 3*x + 5*x^2 + 1*x^3 + 3*x^4 + 1*x^5 + 1*x^6) + N(x)/(1-x^16)^2 , with N(x) = x^9*((1+x^30) + x*(1+x^28)  + 3*x^2*(1+x^26) + x^3*(1+x^24) + 5*x^4*(1+x^22) + 3*x^5*(1+x^20) + 7*x^6*(1+x^18) + x^7*(1+x^16) + 9*x^8*(1+x^14) + 5*x^9*(1+x^12) + 11*x^10*(1+x^10) + 3*x^11*(1+x^8) + 13*x^12*(1+x^6) + 7*x^13*(1+x^4) + 15*x^14*(1+x^2)+x^15).

a(n+32)-2*a(n+16)+a(n) = 0 for n >= 8.

a(n+8) = A106617(n). - Peter Bala, Feb 28 2019

MAPLE

f:= n -> numer(abs(n-8)/(2*n)):

map(f, [$1..100]); # Robert Israel, Dec 06 2018

MATHEMATICA

a[n_] := Numerator[Abs[n-8]/(2n)]; Array[a, 50] (* Amiram Eldar, Dec 06 2018 *)

CROSSREFS

Cf. A127672 (R), A187360 (C), A232625 (b), A055034 (delta), A093819 (degree if k=1), A232626(degree if k=2), A106609 (k=1, p), A225975 (k=1, q), A106617.

Sequence in context: A030710 A117037 A011476 * A021140 A171036 A304658

Adjacent sequences:  A231187 A231188 A231189 * A231191 A231192 A231193

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Dec 12 2013

STATUS

approved

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Last modified January 27 10:53 EST 2022. Contains 350607 sequences. (Running on oeis4.)