A225975 Square root of A226008(n).

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, 26, 54, 7, 58, 30, 62, 16, 66, 34, 70, 9, 74, 38, 78, 20, 82, 42, 86, 11, 90, 46, 94, 24, 98, 50, 102, 13, 106, 54, 110, 28, 114, 58

Offset

0,2

Comments

Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).

From Wolfdieter Lang, Dec 04 2013: (Start)

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,

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)

Links

Table of n, a(n) for n=0..58.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

Formula

a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.

a(n) = 2*a(n-8) -a(n-16).

a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.

a(4n) = A022998(n).

a(4n+1) = A017089(n).

a(4n+2) = A016825(n).

a(4n+3) = A017137(n).

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]

From Wolfdieter Lang, Dec 04 2013: (Start)

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..

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)

Example

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.

Mathematica

a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, May 30 2013 *)
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 *)

Crossrefs

Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.

Sequence in context: A010245 A154196 A248617 * A016529 A077894 A053214

Adjacent sequences: A225972 A225973 A225974 * A225976 A225977 A225978

Keyword

nonn,easy

Author

Paul Curtz, May 22 2013

Extensions

Edited by Bruno Berselli, May 24 2013

Status

approved