A164116 -id:A164116 - cp's OEIS frontend

A049347 Period 3: repeat [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

Wolfdieter Lang

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008
Hankel transform of A099324. - Paul Barry, Aug 10 2009
A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012
a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014
The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

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

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

&cat[[1,-1,0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

A049347 := proc(n)
    op(modp(n,3)+1,[1,-1,0]) ;
end proc:
seq(A049347(n),n=0..100) ; # R. J. Mathar, Aug 06 2016

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)
CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

{a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

G.f.: 1/(1+x+x^2).
a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.
a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)
a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004
a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).
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. - Michael Somos, Oct 03 2006
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006
a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006
From Michael Somos, Oct 03 2006: (Start)
G.f.: (1 - x) /(1 - x^3).
a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)
From Michael Somos, Apr 29 2012: (Start)
G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).
a(n) = (-1)^n * A010892(n).
a(n) * n! = A194770(n+1).
Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). (End)
a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008
G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012
a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013
a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014
a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Edited by Charles R Greathouse IV, Mar 23 2010

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

Original entry on oeis.org

1, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1

Offset: 0

Paul Barry, Oct 27 2004

A transform of (-1)^n.
Row sums of Riordan array ((1-x)/(1+x), x/(1+x)^2), A110162.
Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)(-1)^(n-2k). Then a(n) = b(n) - b(n-2) = A049347(n) - A049347(n-2) (n > 0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.
A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i, n >= 0, with i = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

			G.f. = 1 - x - x^2 + 2*x^3 - x^4 - x^5 + 2*x^6 - x^7 - x^8 + 2*x^9 - x^10 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A061347, A100051, A100063, A098554, A110162.

A099837 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[1,-1,-1]) ;
    else
        -procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A099837(n),n=0..80) ; # R. J. Mathar, Apr 26 2022

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-François Alcover, Feb 15 2012, after Michael Somos *)
LinearRecurrence[{-1, -1}, {1, -1, -1}, 50] (* G. C. Greubel, Aug 08 2017 *)

A099837(n) := block(
        if n = 0 then 1 else [2,-1,-1][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

Vec((1-x^2)/(1+x+x^2) + O(x^20)) \\ Felix Fröhlich, Aug 08 2017

G.f.: (1-x^2)/(1+x+x^2).
Euler transform of length 3 sequence [-1, -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 3 sequence [-1, 0, 3]. - Michael Somos, Mar 22 2011
a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e > 0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012
a(n) = a(-n). a(n) = c_3(n) if n > 1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n = 0, 1, 2. - Michael Somos, Jan 19 2012
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011
a(n+3) = R(n,-1) for n >= 0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014
For n > 0, a(n) = 2*cos(n*Pi/3)*cos(n*Pi). - Wesley Ivan Hurt, Sep 25 2017
From Peter Bala, Apr 20 2024: (Start)
a(n) is equal to the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. Cf. A333093.
Row sums of the Riordan array A110162. (End)

A080891 Period 5: repeat [0, 1, -1, -1, 1].

Original entry on oeis.org

0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0

Offset: 0

N. J. A. Sloane, Sep 23 2003

a(n) = (5/n), where (k/n) is the Kronecker symbol.
L(1;5) (Dirichlet L-series) is the integral from 0 to 1 of the g.f. of a(n+1). Partial sums are A092202. - Paul Barry, Apr 01 2005
From R. J. Mathar, Jul 15 2010, simplified Jul 27 2010: (Start)
The sequence is the real non-principal Dirichlet character mod 5. (The principal character mod 5 is A011558.)
Associated Dirichlet L-functions are, for example, L(1,chi) = Sum_{n>=1} a(n)/n = A086466 or L(2,chi) = Sum_{n>=1} a(n)/n^2 = 0.7062114... = 4*Pi^2/(25*sqrt(5)). (End)
This sequence {a(n)} appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + a(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), C(n) = A156174(n+4) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116. - Wolfdieter Lang, Feb 26 2014
In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 4}(x). - Michael Somos, Sep 04 2015

			G.f. = x - x^2 - x^3 + x^4 + x^6 - x^7 - x^8 + x^9 + x^11 - x^12 - x^13 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=5, Chi_2(n).
H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1962, p. 173.

J. B. Gil and S. Robins, Hecke operators on rational functions, arXiv:math/0309244 [math.NT], 2003.
J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 435.
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).
Index entries for two-way infinite sequences

Cf. A011558, A086937, A100047.

&cat [[0, 1, -1, -1, 1]^^30]; // Wesley Ivan Hurt, Dec 26 2016

A080891 := proc(n) numtheory[jacobi](n,5) ; end proc: seq(A080891(n),n=0..100) ; # R. J. Mathar, Jul 29 2010

a[ n_] := Mod[n^2 + 1, 5] - 1; (* Michael Somos, May 24 2015 *)
a[ n_] := KroneckerSymbol[ n, 5]; (* Michael Somos, May 24 2015 *)
a[ n_] := {1, -1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* Michael Somos, May 24 2015 *)
PadRight[{},120,{0,1,-1,-1,1}] (* Harvey P. Dale, Nov 30 2023 *)

numlib::jacobi(n,5)$ n=0..100 // Zerinvary Lajos, May 13 2008

a(n)=kronecker(5,n) /* Also, a(n)=kronecker(n,5) */

{a(n) = (n^2 + 1)%5 - 1}; /* Michael Somos, Dec 01 2004 */

If n == 0 (mod 5) a(n)=0; if n == 1 or 4 (mod 5) a(n)=1; if n == 2 or 3 (mod 5) a(n)=-1.
G.f.: x*(1-x^2)/(1+x+x^2+x^3+x^4). - Paul Barry, Apr 01 2005
G.f.: x * (1 - x) * (1 - x^2) / (1 - x^5). a(n) = a(-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2005
Euler transform of length 5 sequence [-1, -1, 0, 0, 1]. - Michael Somos, Jun 17 2005
Transform of the Fibonacci numbers by the Riordan array A102587. - Paul Barry, Jul 14 2005
a(n) = -1 + floor(12002/99999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
a(n) = -1 + floor(137/242*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 04 2013
|A011558(n)| = |a(n)| = |A100047(n)|. - Michael Somos, May 24 2015
a(n) is completely multiplicative with a(p) = Kronecker(5, p). - Michael Somos, Jun 17 2015
From Wesley Ivan Hurt, Dec 26 2016: (Start)
a(n) = a(n-5) for n > 4.
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 3.
a(n) = 1 + 2*floor((n-4)/5) - 2*floor((n-2)/5) + floor((n-1)/5) - floor(n/5). (End)
a(n) = 2*(cos(2*n*Pi/5) - cos(4*n*Pi/5))/sqrt(5). - Wesley Ivan Hurt, Sep 26 2018
a(n) = a(n-1)*a(n-4) - a(n-2)*a(n-3) for n > 3. - Nicolas Bělohoubek, May 21 2024
a(n) = n^2 - 5*floor((n^2+1)/5). - Aaron J Grech, Aug 28 2024

Name specified by Wolfdieter Lang, Feb 26 2014

A010891 Inverse of 5th cyclotomic polynomial.

Original entry on oeis.org

1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1

Offset: 0

Simon Plouffe

D(n):= a(n+3) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and C(n) = A156174(n+4) for n >= 0. See one of the comments on A164116. - Wolfdieter Lang, Feb 26 2014

Periodic with period length 5. - Ray Chandler, Apr 03 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).
Index to sequences related to inverse of cyclotomic polynomials

&cat[[1,-1,0,0,0]: n in [0..20]]; // Vincenzo Librandi, Apr 03 2014

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
# alternative
A010891 := proc(n)
    op(1+(n mod 5),[1,-1,0,0,0]) ;
end proc:
seq(A010891(n),n=0..20) ; # R. J. Mathar, Feb 27 2025

CoefficientList[Series[1/Cyclotomic[5, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

Vec(1/polcyclo(5)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

G.f.: 1/ ( 1+x+x^2+x^3+x^4 ). - R. J. Mathar, Mar 11 2011

A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

Original entry on oeis.org

1, 2, 5, 9, 15, 21, 29, 38, 48, 60, 72, 86, 101, 117, 135, 153, 173, 194, 216, 240, 264, 290, 317, 345, 375, 405, 437, 470, 504, 540, 576, 614, 653, 693, 735, 777, 821, 866, 912, 960, 1008, 1058, 1109, 1161, 1215, 1269, 1325, 1382, 1440, 1500, 1560, 1622, 1685, 1749, 1815, 1881, 1949, 2018, 2088, 2160, 2232, 2306, 2381

Offset: 1

Clark Kimberling, Jan 16 2011

Apart from the initial term this is the elliptic troublemaker sequence R_n(2,5) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences see the cross references below. - Peter Bala, Aug 08 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT]
Wikipedia, Maximum number of deciamonds in a hexagon.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 1, -2, 1).

Cf. A183532, A279169.

Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A184535 (= R_n(2,5) = R_n(3,5)).

Concatenation([1], List([2..10^3], n->Int(3/5 * n^2))); # Muniru A Asiru, Feb 04 2018

1,seq(floor(3/5*n^2), n=2..10^3); # Muniru A Asiru, Feb 04 2018

p[n_] := FractionalPart[(n^3 + 5)^(1/3)]; q[n_] := Floor[1/p[n]]; Table[q[n], {n, 1, 120}]
Join[{1},LinearRecurrence[{2, -1, 0, 0, 1, -2, 1},{2, 5, 9, 15, 21, 29, 38},62]] (* Ray Chandler, Aug 31 2015 *)

a(n) = if(n==1, 1, 3*n^2\5); \\ Altug Alkan, Mar 03 2018

def A184535(n): return 3*n**2//5 if n>1 else 1 # Chai Wah Wu, Aug 04 2025

a(n) = floor(1/{(5+n^3)^(1/3)}), where {}=fractional part.
a(n)= +2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7), for n>8, with g.f. 1-x^2*(1+x)*(2*x^2-x+2)/ ((x^4+x^3+x^2+x+1) *(x-1)^3), so a(n) is (3n^2-2)/5 plus a fifth of A164116 for n>1. [Bruno Berselli, Jan 30 2011. See the following Bala's comment.]
From Peter Bala, Aug 08 2013: (Start)
a(n) = floor(3/5*n^2) for n >= 2.
The sequence b(n) := floor(3/5*n^2) - 3/5*n^2, n >= 1, is periodic with period [-3/5, -2/5, -2/5, -3/5, 0] of length 5. The generating function and recurrence equation given above easily follow from these observations.
The sequence c(n) := 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ) is also periodic with period 5, and calculation shows it has the same period as the sequence b(n). Thus b(n) = c(n), yielding the alternative formula a(n) = 3/5*n^2 + 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ), which is one of the formulas for the elliptic troublemaker sequence R_n(2,5) given in Stange (see Section 7, equation (21)). (End)

Better name from Peter Bala, Aug 08 2013

A156174 Period 5: repeat [1,-1,1,-1,0].

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0

Offset: 0

N. J. A. Sloane, Nov 06 2009

C(n) := a(n+4) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116(n+5). - Wolfdieter Lang, Feb 26 2014

With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 1, z = 1. - Michael Somos, Oct 17 2018

			G.f. = 1 - x + x^2 - x^3 + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Cf. A010874, A011558 (this read mod 2), A099443, A198517.

A156174:=n->[1,-1,1,-1,0][(n mod 5)+1]: seq(A156174(n), n=0..100); # Wesley Ivan Hurt, May 31 2015

CoefficientList[Series[(1 + x^2)/(1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* Wesley Ivan Hurt, May 31 2015 *)
a[ n_] := { -1, 1, -1, 0, 1}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := (-1)^Mod[n, 5] Sign @ Mod[n + 1, 5]; (* Michael Somos, Jun 17 2015 *)

a(n)=[1,-1,1,-1,0][n%5+1] \\ Charles R Greathouse IV, Oct 28 2011

{a(n) = (-1)^(n%5) * sign((n+1)%5)}; /* Michael Somos, Jun 17 2015 */

G.f.: (1+x^2)/(1 + x + x^2 + x^3 + x^4).
Sum_{i=0..n} a(i) = A198517(n). - Bruno Berselli, Nov 02 2011
From Wesley Ivan Hurt, May 31 2015: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 4.
a(n) = Sum_{i=0..3} A011558(n+2+i)*(-1)^i. (End)
Euler transform of length 5 sequence [-1, 1, 0, -1, 1]. - Michael Somos, Jun 17 2015
G.f.: (1-x)*(1-x^4)/((1-x^2)*(1-x^5)). - Michael Somos, Jun 17 2015
a(n) = -a(-2-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2015
a(n) = (2/5) * (cos(4*(n-2)*Pi/5) + cos(2*n*Pi/5) + cos(4*n*Pi/5) - cos(2*(n-3)*Pi/5) - cos(4*(n-3)*Pi/5) - cos(2*(n-1)*Pi/5) - cos(4*(n-1)*Pi/5) - cos((2*n+1)*Pi/5)). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (-1)^n * A099443(n). - Michael Somos, Oct 17 2018
a(5*n) = a(5*n + 2) = 1, a(5*n + 1) = a(5*n + 3) = -1, a(5*n + 4) = 0 for all n in Z. - Michael Somos, Nov 27 2019

A234044 Period 7: repeat [2, -2, 1, 0, 0, 1, -2].

Original entry on oeis.org

2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2

Offset: 0

Wolfdieter Lang, Feb 27 2014

This is a member of the six sequences which appear for the instance N=7 of the general formula 2*exp(2*Pi*n*I/N) = R(n, x^2-2) + x*S(n-1, x^2-2)*s(N)*I, for n >= 0, with I = sqrt(-1), s(N) = sqrt(2-x)*sqrt(2+x), x = rho(N) := 2*cos(Pi/N) and R and S are the monic Chebyshev polynomials whose coefficient tables are given in A127672 and A049310. If powers x^k with k >= delta(N) = A055034(N) enter in R or x*S then C(N, x), the minimal polynomial of x = rho(N) (see A187360) is used for a reduction. If delta(N) = 2 it may happen that sqrt(2+x) or sqrt(2-x) is an integer in the number field Q(rho(N)). See the N=5 case comment on A164116.

For N=7 with delta(7) = 3, and C(7, x) = x^3 - x^2 - 2*x + 1 the final result becomes 2*exp(2*Pi*n*I/7) = (a(n) + b(n)*x + c(n)*x^2) + (A(n) + B(n)*x + C(n)*x^2)*s(7)*I, with x = rho(7) = 2*cos(Pi/7), a(n) the present sequence, b(n) = A234045(n), c(n) = A234046(n), A(n) = A238468(n), B(n) = A238469(n) and C(n) = A238470(n). The a, b, c and A, B, C brackets are integers in Q(rho(7)).

			n = 4: 2*exp(8*Pi*I/7) = (2-16*x^2+20*x^4-8*x^6+x^8) + (4*x+10*x^3-6*x^5+x^7)*s(7)*I, reduced with C(7, x) = x^3 - x^2 - 2*x + 1 = 0 this becomes = (-x) + (-1)*s(7)*I with x= 2*cos(Pi/7) and s(7) = 2*sin(Pi/7).The power basis coefficients are thus (a(4), b(4), c(4)) = (0, -1, 0) and (A(4), B(4), C(4)) = (-1, 0, 0).

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

Cf. A234045, A234046, A238468, A238469, A238470, A099837 (N=3), A056594 (N=4), A164116 (N=5), A057079 (N=6).

&cat [[2, -2, 1, 0, 0, 1, -2]^^20]; // Wesley Ivan Hurt, Jul 16 2016

seq(op([2, -2, 1, 0, 0, 1, -2]), n=0..20); # Wesley Ivan Hurt, Jul 16 2016

PadRight[{}, 100, {2, -2, 1, 0, 0, 1, -2}] (* Wesley Ivan Hurt, Jul 16 2016 *)

a(n)=[2, -2, 1, 0, 0, 1, -2][n%7+1] \\ Charles R Greathouse IV, Jul 17 2016

G.f.: (2 - 2*x + x^2 + x^5 - 2*x^6)/(1 - x^7).
a(n+7) = a(n) for n>=0, with a(0) = -a(1) = -a(6) = 2, a(3) = a(4) =0 and a(2) = a(5) = 1.
From Wesley Ivan Hurt, Jul 16 2016: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) = 0 for n>5.
a(n) = (1/7) * Sum_{k=1..6} 2*cos((2k)*n*Pi/7) - 2*cos((2k)*(1+n)*Pi/7) + cos((2k)*(2+n)*Pi/7) + cos((2k)*(5+n)*Pi/7) - 2*cos((2k)*(6+n)*Pi/7).
a(n) = 2 + 4*floor(n/7) - 3*floor((1+n)/7) + floor((2+n)/7) - floor((4+n)/7) + 3*floor((5+n)/7) - 4*floor((6+n)/7). (End)

A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Michael Somos, Aug 10 2009

The sequence A107453 has the same terms but different offset.
Convolution inverse of A164116.
Decimal expansion of 11111/99990. - Elmo R. Oliveira, Feb 18 2024

			1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...

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

Cf. A107453, A138191, A164116, A164117.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 22 2018 *)
LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* Harvey P. Dale, Aug 24 2019 *)

PARI
```
{a(n) = 2 - (n==0) - (n%4>0)}
```

x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ Altug Alkan, Sep 23 2018

Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].
a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.
a(n) = (-1)^n * A164117(n).
a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.
a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.
G.f.: (1 + x + x^2 + x^3 + x^4) / ((1+x)*(1-x)*(1+x^2)).
a(n) = A138191(n+2), n>0. - R. J. Mathar, Aug 17 2009
Dirichlet g.f. (1+1/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - Bruno Berselli, Feb 25 2011

A164118 Expansion of (1 - x^2) * (1 - x^4) * (1 - x^5) / ((1 - x) * (1 - x^10)) in powers of x.

Original entry on oeis.org

1, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1

Offset: 0

Michael Somos, Aug 10 2009

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

A164116(n) = (-1)^n * a(n). Convolution inverse of A164117.

CoefficientList[Series[(1 - x^4) / (1 - x + x^2 - x^3 + x^4), {x, 0, 50}], x] (* G. C. Greubel, Sep 12 2017 *)

{a(n) = -(n==0) + [2, 1, 0, 0, -1, -2, -1, 0, 0, 1][n%10 + 1]}

Euler transform of length 10 sequence [ 1, -1, 0, -1, -1, 0, 0, 0, 0, 1].
a(-n) = a(n). a(n + 5) = -a(n) unless n=0 or n=-5.
G.f.: (1 - x^4) / (1 - x + x^2 - x^3 + x^4).

A257181 Expansion of (1 - x) * (1 + x^4) / (1 + x^5) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0

Offset: 0

Michael Somos, Apr 17 2015

			G.f. = 1 - x + x^4 - 2*x^5 + x^6 - x^9 + 2*x^10 - x^11 + x^14 - 2*x^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,-1).

Cf. A164116, A257179.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1+x^4)/(1+x^5))); // G. C. Greubel, Aug 02 2018

a[ n_] := -Boole[n == 0] + {-1, 0, 0, 1, -2, 1, 0, 0, -1, 2}[[Mod[n, 10, 1]]];
a[ n_] := SeriesCoefficient[ (1 - x) * (1 + x^4) / (1 + x^5), {x, 0, Abs@n}];
CoefficientList[Series[(1-x)*(1+x^4)/(1+x^5), {x, 0, nmax}], x] (* G. C. Greubel, Aug 02 2018 *)

{a(n) = if( n==0, 1, (-1)^(n\5) * [2, -1, 0, 0, 1][n%5 + 1])};

{a(n) = polcoeff( (1 - x) * (1 + x^4) / (1 + x^5) + x * O(x^abs(n)), abs(n))};

x='x+O('x^60); Vec((1-x)*(1+x^4)/(1+x^5)) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 10 sequence [-1, 0, 0, 1, -1, 0, 0, -1, 0, 1].
a(n) = a(-n) for all n in Z. a(n+5) = -a(n) unless n = 0 or -5. a(5*n) = 2 * (-1)^n unless n = 0. a(5*n + 2) = a(5*n + 3) = 0. a(5*n + 1) = a(5*n - 1) = -(-1)^n.
G.f.: (1 - x) * (1 + x^4) / (1 + x^5).
G.f.: (1 - x) * (1 - x^5) * (1 - x^8) / ((1 - x^4) * (1 - x^10)).
Convolution inverse is A257179.
a(n) = (-1)^floor( (n+4) / 5) * A164116(n).

cp's OEIS Frontend

A049347 Period 3: repeat [1, -1, 0].

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A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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A080891 Period 5: repeat [0, 1, -1, -1, 1].

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A010891 Inverse of 5th cyclotomic polynomial.

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A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

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