A086227 a(n) = Sum_{1<=k<=4*n, gcd(k,n)=1} (i^k*tan(k*Pi/(4*n)))/(4*i), where i is the imaginary unit.

-1, 2, -2, 2, -4, 4, -4, 6, -4, 6, -8, 6, -8, 8, -8, 8, -12, 10, -8, 16, -12, 12, -16, 10, -12, 18, -16, 14, -16, 16, -16, 24, -16, 16, -24, 18, -20, 24, -16, 20, -32, 22, -24, 24, -24, 24, -32, 28, -20, 32, -24, 26, -36, 24, -32, 40, -28, 30, -32, 30, -32, 48, -32, 24, -48, 34, -32, 48, -32, 36, -48, 36, -36, 40, -40, 48, -48

Offset: 2

Author

Benoit Cloitre, Aug 28 2003

Keywords

• sign

Comments

This seems to be (-1)^(n+1) times h(-4n^2) = (-1)^(n+1)*A000003(n^2), where h(k) is the class number. Verified for n <= 10^5. - Charles R Greathouse IV, Apr 28 2013

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10000
• Stanley Rabinowitz, Problem 2129, Crux Mathematicorum, Vol. 22, No. 3 (1996), p. 123; Solution to Problem 2129, by G. P. Henderson and Kurt Girstmair, ibid., Vol. 23, No. 4 (1997), pp. 246-249.

Crossrefs

Cf. A000003, A204617,A079458, A140434.

Programs

Mathematica

f[p_, e_] := p^(e - 1) * Switch[Mod[p, 4], 2, 1, 1, p - 1, 3, p + 1]; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := If[EvenQ[n], -s[n], s[n]/2]; Array[a, 100, 2] (* Amiram Eldar, Mar 07 2022 *)

PARI

a(n)=round(real(1/4/I*sum(k=1,4*n,(I^k)*tan(Pi/4/n*if(gcd(k,n)-1,0,k)))))

PARI

a(n)=round(imag(sum(k=1,4*n,if(gcd(k,n)==1,I^k*tan(k*Pi/4/n))))/4) \\ Charles R Greathouse IV, Apr 25 2013

PARI

a(n)=my(s);for(k=1,2*n,if(gcd(2*k-1,n)==1,s-=(-1)^k*tan((2*k-1)*Pi/4/n))); round(s/4) \\ Charles R Greathouse IV, Apr 25 2013

Formula

a(n) = -A204617(n) if n is even, and A204617(n)/2 if n is odd (Rabinowitz, 1996). - Amiram Eldar, Mar 07 2022

a(n) = (-1)^(n+1)*A079458(n)/A140434(n). - Ridouane Oudra, Jun 23 2024

Extensions

Definition corrected by Charles R Greathouse IV, Apr 25 2013