A232625 Denominators of abs(n-8)/(2*n), n >= 1.

2, 2, 6, 2, 10, 6, 14, 1, 18, 10, 22, 6, 26, 14, 30, 4, 34, 18, 38, 10, 42, 22, 46, 3, 50, 26, 54, 14, 58, 30, 62, 8, 66, 34, 70, 18, 74, 38, 78, 5, 82, 42, 86, 22, 90, 46, 94, 12, 98, 50, 102, 26, 106, 54, 110, 7, 114, 58, 118, 30, 122, 62, 126, 16, 130, 66

Offset: 1

Author

Wolfdieter Lang, Dec 12 2013

Keywords

• nonn
• easy

Comments

The numerators are given in A231190. See the comments there on 2*sin(Pi*4/n).

2*sin(Pi*4/n) = R(b(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/a(n)) =: rho(a(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively. b(n) = A231190(n).

delta(a(n)) = deg(2,n), with delta(k) = A055034(k), is the degree of the algebraic number 2*sin(Pi*4/n) given in A232626.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A127672 (R), A187360 (C), A231190 (b), A055034 (delta), A232626 (degree k=2), A106609 (k=1, p), A225975 (k=1, q), A093819 (degree k=1).

Programs

Mathematica

a[n_] := Denominator[(n-8)/(2*n)]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)

PARI

a(n) = denominator((n-8)/(2*n)); \\ Amiram Eldar, Nov 09 2024

Formula

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

a(n) = 2*n/gcd(n-8, 16).

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

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

Sum_{k=1..n} a(k) ~ (171/256) * n^2. - Amiram Eldar, Nov 09 2024