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

%I A232625 #19 Jun 02 2025 08:50:42
%S A232625 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,
%T A232625 54,14,58,30,62,8,66,34,70,18,74,38,78,5,82,42,86,22,90,46,94,12,98,
%U A232625 50,102,26,106,54,110,7,114,58,118,30,122,62,126,16,130,66
%N A232625 Denominators of abs(n-8)/(2*n), n >= 1.
%C A232625 The numerators are given in A231190. See the comments there on 2*sin(Pi*4/n).
%C A232625 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).
%C A232625 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.
%H A232625 Amiram Eldar, <a href="/A232625/b232625.txt">Table of n, a(n) for n = 1..10000</a>
%F A232625 a(n) = denominator(abs(n-8)/(2*n)), n >= 1.
%F A232625 a(n) = 2*n/gcd(n-8, 16).
%F A232625 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).
%F A232625 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.
%F A232625 Sum_{k=1..n} a(k) ~ (171/256) * n^2. - _Amiram Eldar_, Nov 09 2024
%t A232625 a[n_] := Denominator[(n-8)/(2*n)]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2024 *)
%o A232625 (PARI) a(n) = denominator((n-8)/(2*n)); \\ _Amiram Eldar_, Nov 09 2024
%Y A232625 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).
%K A232625 nonn,easy
%O A232625 1,1
%A A232625 _Wolfdieter Lang_, Dec 12 2013