cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A010892 Inverse of 6th cyclotomic polynomial. A period 6 sequence.

Original entry on oeis.org

1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0
Offset: 0

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Author

Simon Plouffe

Keywords

Comments

Any sequence b(n) satisfying the recurrence b(n) = b(n-1) - b(n-2) can be written as b(n) = b(0)*a(n) + (b(1)-b(0))*a(n-1).
a(n) is the determinant of the n X n matrix M with m(i,j)=1 if |i-j| <= 1 and 0 otherwise. - Mario Catalani (mario.catalani(AT)unito.it), Jan 25 2003
Also row sums of triangle in A108299; a(n)=L(n-1,1), where L is also defined as in A108299; see A061347 for L(n,-1). - Reinhard Zumkeller, Jun 01 2005
Pisano period lengths: 1, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ... - R. J. Mathar, Aug 10 2012
Periodic sequences of this type can also be calculated as a(n) = c + floor(q/(p^m-1)*p^n) mod p, where c is a constant, q is the number representing the periodic digit pattern and m is the period. c, p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D, min = minimum value of elements in D. Then c := min, p := max - min + 1 and q := p^m*Sum_{i=1..m} (D(i)-min)/p^i. Example: D = (1, 1, 0, -1, -1, 0), c = -1, m = 6, p = 3 and q = 676 for this sequence. - Hieronymus Fischer, Jan 04 2013
B(n) = a(n+5) = S(n-1, 1) appears, together with a(n) = A057079(n+1), in the formula 2*exp(Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i with i = sqrt(-1). For S(n, x) see A049310. See also a Feb 27 2014 comment on A099837. - Wolfdieter Lang, Feb 27 2014
a(n) (for n>=1) is the difference between numbers of even and odd permutations p of 1,2,...,n such that |p(i)-i|<=1 for i=1,2,...,n. - Dmitry Efimov, Jan 08 2016
From Tom Copeland, Jan 31 2016: (Start)
Specialization of the o.g.f. 1 / ((x - w1)(x-w2)) = (1/(w1-w2)) ((w1-w2) + (w1^2 - w2^2) x + (w1^3-w2^3) x^2 + ...) with w1*w2 = (1/w1) + (1/w2) = 1. Then w1 = q = e^(i*Pi/3) and w2 = 1/q = e^(-i*Pi/3), giving the o.g.f. 1 /(1-x+x^2) for this entry with a(n) = (2/sqrt(3)) sin((n+1)Pi/3). See the Copeland link for more relations.
a(n) = (q^(n+1) - q^(-(n+1))) / (q - q^(-1)), so this entry gives the o.g.f. for an instance of the quantum integers denoted by [m]_q in Morrison et al. and Tingley. (End)

Examples

			G.f. = 1 + x - x^3 - x^4 + x^6 + x^7 - x^9 - x^10 + x^12 + x^13 - x^15 + ...

Links

Crossrefs

a(n) = row sums of signed triangle A049310.
Cf. A049347, A057078.
Differs only by a shift from A128834.
a(n+1) = row sums of triangle A130777: repeat(1,0,-1,-1,0,1).

Programs

  • Magma
    &cat[[1,1,0,-1,-1,0]: n in [0..20]]; // Vincenzo Librandi, Apr 03 2014
  • Maple
    a:=n->coeftayl(1/(x^2-x+1), x=0, n);
a:=n->2*sin(Pi*(n+1)/3)/sqrt(3);
A010892:=n->[1,1,0,-1,-1,0][irem(n,6)+1];
A010892:=n->Array(0..5,[1,1,0,-1,-1,0])[irem(n,6)];
A010892:=n->table([0=1,1=1,2=0,3=-1,4=-1,5=0])[irem(n,6)];
with(numtheory,cyclotomic); c := series(1/cyclotomic(6,x),x,102): seq(coeff(c,x,n),n=0..101); # Rainer Rosenthal, Jan 01 2007
  • Mathematica
    a[n_] := {1, 1, 0, -1, -1, 0}[[Mod[n, 6] + 1]]; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Jul 19 2013 *)
CoefficientList[Series[1/Cyclotomic[6, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
PadRight[{},120,{1,1,0,-1,-1,0}] (* Harvey P. Dale, Jul 07 2020 *)
  • PARI
    {a(n) = (-1)^(n\3) * sign((n + 1)%3)}; /* Michael Somos, Sep 23 2005 */
  • PARI
    {a(n) = subst( poltchebi(n) + poltchebi(n-1), 'x, 1/2) * 2/3}; /* Michael Somos, Sep 23 2005 */
  • PARI
    {a(n) = [1, 1, 0, -1, -1, 0][n%6 + 1]}; /* Michael Somos, Feb 14 2006 */
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n++; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, 0, p%6 == 1, 1, (-1)^e)))}; /* Michael Somos, Oct 29 2006 */
  • Python
    def A010892(n): return [1,1,0,-1,-1,0][n%6] # Alec Mihailovs, Jan 01 2007
  • Sage
    [lucas_number1(n,1,+1) for n in range(-5, 97)] # Zerinvary Lajos, Apr 22 2009
  • Sage
    def A010892():
    x, y = -1, -1
    while True:
        yield -x
        x, y = y, -x + y
a = A010892()
[next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

Formula

G.f.: 1 / (1 - x + x^2).
a(n) = a(n-1) - a(n-2), a(0)=1, a(1)=1.
a(n) = ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2.
a(n) = 0 if n mod 6 = 2 or 5, a(n) = +1 if n mod 6 = 0 or 1, a(n) = -1 otherwise. a(n) = S(n, 1) = U(n, 1/2) (Chebyshev U(n, x) polynomials).
a(n) = sqrt(4/3)*Im((1/2 + i*sqrt(3/4))^(n+1)). - Henry Bottomley, Apr 12 2000
Binomial transform of A057078. a(n) = Sum_{k=0..n} C(k, n-k)*(-1)^(n-k). - Paul Barry, Sep 13 2003
a(n) = 2*sin(Pi*n/3 + Pi/3)/sqrt(3). - Paul Barry, Jan 28 2004
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k. - Paul Barry, Jul 28 2004
Euler transform of length 6 sequence [1, -1, -1, 0, 0, 1]. - Michael Somos, Sep 23 2005
a(n) = a(1 - n) = -a(-2 - n) for all n in Z. - Michael Somos, Feb 14 2006
a(n) = Sum_{k=0..n} (-2)^(n-k) * A085838(n,k). - Philippe Deléham, Oct 26 2006
a(n) = b(n+1) where b(n) is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 6), b(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Oct 29 2006
Given g.f. A(x), then, B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v - 2*u*v * (1 - u). - Michael Somos, Oct 29 2006
a(2*n) = A057078(n), a(2*n+1) = A049347(n).
a(n) = Sum_{k=0..n} A109466(n,k). - Philippe Deléham, Nov 14 2006
a(n) = Sum_{k=0..n} A133607(n,k). - Philippe Deléham, Dec 30 2007
a(n) = A128834(n+1). - Jaume Oliver Lafont, Dec 05 2008
a(n) = Sum_{k=0..n} C(n+k+1,2k+1) * (-1)^k. - Paul Barry, Jun 03 2009
a(n) = A101950(n,0) = (-1)^n * A049347(n). - Philippe Deléham, Feb 10 2012
a(n) = Product_{k=1..floor(n/2)} 1 - 4*(cos(k*Pi/(n+1)))^2. - Mircea Merca, Apr 01 2012
G.f.: 1 / (1 - x / (1 + x / (1 - x))). - Michael Somos, Apr 02 2012
a(n) = -1 + floor(181/819*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(13/14*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 04 2013
a(n) = 1/(1+r2)*(1/r1)^n + 1/(1+r1)*(1/r2)^n, with r1=(1-i*sqrt(3))/2 and r2=(1+i*sqrt(3))/2. - Ralf Stephan, Jul 19 2013
a(n) = ((n+1)^2 mod 3) * (-1)^floor((n+1)/3). - Wesley Ivan Hurt, Mar 15 2015
a(n-1) = n - Sum_{i=1..n-1} i*a(n-i). - Derek Orr, Apr 28 2015
a(n) = S(2*n+1, sqrt(3))/sqrt(3) = S(n, 1) with S(n, x) coefficients given in A049310. The S(n, 1) formula appeared already above. S(2*n, sqrt(3)) = A057079(n). See also a Feb 27 2014 comment above. - Wolfdieter Lang, Jan 16 2018
E.g.f.: sqrt(exp(x)*4/3) * cos(x*sqrt(3/4) - Pi/6). - Michael Somos, Jul 05 2018
a(n) = Determinant(Tri(n)), for n >= 1, with Tri(n) the n X n tridiagonal matrix with entries 1 (a special Toeplitz matrix). - Wolfdieter Lang, Sep 20 2019
a(n) = Product_{k=1..n}(1 + 2*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

Extensions

Entry revised by N. J. A. Sloane, Jul 16 2004

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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