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%I #40 Aug 09 2022 14:13:37
%S 0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30,8,34,18,38,5,42,22,46,12,50,
%T 26,54,7,58,30,62,16,66,34,70,9,74,38,78,20,82,42,86,11,90,46,94,24,
%U 98,50,102,13,106,54,110,28,114,58
%N Square root of A226008(n).
%C Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).
%C From _Wolfdieter Lang_, Dec 04 2013: (Start)
%C This sequence a(n), n>=1, appears in the formula 2*sin(2*Pi/n) = R(p(n), x) modulo C(a(n), x), with x = rho(a(n)) = 2*cos(Pi/a(n)), the R-polynomials given in A127672 and the minimal C-polynomials of rho given in A187360. This follows from the identity 2*sin(2*Pi/n) = 2*cos(Pi*p(n)/a(n)) with gcd(p(n), a(n)) = 1. For p(n) see a comment on A106609,
%C Because R is an integer polynomial it shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(a(n))) of degree delta(a(n)) (the degree of C(a(n), x)), with delta(k) = A055034(k). This degree is given in A093819. For the coefficients of 2*sin(2*Pi/n) in the power basis of Q(rho(a(n))) see A231189 . (End)
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).
%F a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.
%F a(n) = 2*a(n-8) -a(n-16).
%F a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.
%F a(4n) = A022998(n).
%F a(4n+1) = A017089(n).
%F a(4n+2) = A016825(n).
%F a(4n+3) = A017137(n).
%F G.f.: x*(2 +2*x +6*x^2 +x^3 +10*x^4 +6*x^5 +14*x^6 +4*x^7 +14*x^8 +6*x^9 +10*x^10 +x^11 +6*x^12 +2*x^13 +2*x^14)/((1-x)^2*(1+x)^2*(1+x^2)^2*(1+x^4)^2). [_Bruno Berselli_, May 23 2013]
%F From _Wolfdieter Lang_, Dec 04 2013: (Start)
%F a(n) = 2*n if n is odd; if n is even then a(n) is n if n/2 == 1, 3, 5, 7 (mod 8), it is n/2 if n/2 == 0, 4 (mod 8) and it is n/4 if n/2 == 2, 6 (mod 8). This leads to the given G.f..
%F With c(n) = A178182(n), n>=1, a(n) = c(n)/2 if c(n) is even and c(n) if c(n) is odd. This leads to the preceding formula. (End)
%e For the first formula: a(0)=-1+1=0, a(1)=-3+5=2, a(2)=-1+3=2, a(3)=-1+7=6, a(4)=0+1=1.
%t a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* _Jean-François Alcover_, May 30 2013 *)
%t LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30},60] (* _Harvey P. Dale_, Nov 21 2019 *)
%Y Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, May 22 2013
%E Edited by _Bruno Berselli_, May 24 2013
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