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A225975 Square root of A226008(n). 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified January 28 22:33 EST 2020. Contains 331328 sequences. (Running on oeis4.)