A057077 Periodic sequence 1,1,-1,-1; expansion of (1+x)/(1+x^2).

1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1

Offset

0,1

Comments

Sum_{k>=0} a(k)/(k+1) = Sum_{k>=0} 1/((a(k)*(k+1))) = log(2)/2 + Pi/4. - Jaume Oliver Lafont, Apr 30 2010

Abscissa of the image produced after n alternating reflections of (1,1) over the x and y axes respectively. Similarly, the ordinate of the image produced after n alternating reflections of (1,1) over the y and x axes respectively. - Wesley Ivan Hurt, Jul 06 2013

Links

Table of n, a(n) for n=0..71.

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.

T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, Theorem 2.5, k=2.

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

Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: (1+x)/(1+x^2).

a(n) = S(n, 0) + S(n-1, 0) = S(2*n, sqrt(2)); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 0)=A056594.

a(n) = (-1)^binomial(n,2) = (-1)^floor(n/2) = 1/2*((n+2) mod 4 - n mod 4). For fixed r = 0,1,2,..., it appears that (-1)^binomial(n,2^r) gives a periodic sequence of period 2^(r+1), the period consisting of a block of 2^r plus ones followed by a block of 2^r minus ones. See A033999 (r = 0), A143621 (r = 2) and A143622 (r = 3). Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(0) = cos(1) + sin(1), E(1) = cos(1) - sin(1) and E(k) is an integral linear combination of E(0) and E(1) (a Dobinski-type relation). Precisely, E(k) = A121867(k) * E(0) - A121868(k) * E(1). See A143623 and A143624 for the decimal expansions of E(0) and E(1) respectively. For a fixed value of r, similar relations hold between the values of the sums E_r(k) := Sum_{n>=0} (-1)^floor(n/r)*n^k/n!, k = 0,1,2,... . For particular cases see A000587 (r = 1) and A143628 (r = 3). - Peter Bala, Aug 28 2008

a(n) = (-1)^A180969(1,n), where the first index in A180969(.,.) is the row index. - Adriano Caroli, Nov 18 2010

a(n) = (-1)^((2*n+(-1)^n-1)/4) = i^((n-1)*n), with i=sqrt(-1). - Bruno Berselli, Dec 27 2010 - Aug 26 2011

Non-simple continued fraction expansion of (3+sqrt(5))/2 = A104457. - R. J. Mathar, Mar 08 2012

E.g.f.: cos(x)*(1 + tan(x)). - Arkadiusz Wesolowski, Aug 31 2012

From Ricardo Soares Vieira, Oct 15 2019: (Start)

E.g.f.: sin(x) + cos(x) = sqrt(2)*sin(x + Pi/4).

a(n) = sqrt(2)*(d^n/dx^n) sin(x)|_x=Pi/4, i.e., a(n) equals sqrt(2) times the n-th derivative of sin(x) evaluated at x=Pi/4. (End)

a(n) = 4*floor(n/4) - 2*floor(n/2) + 1. - Ridouane Oudra, Mar 23 2024

Maple

seq((-1)^floor(k/2), k=0..70); # Wesley Ivan Hurt, Jul 06 2013

Mathematica

a[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]] (* Jean-François Alcover, Jul 05 2013 *)
PadRight[{}, 80, {1, 1, -1, -1}] (* Harvey P. Dale, Jun 21 2015 *)

Prog

(Maxima) A057077(n) := block(
        [1, 1, -1, -1][1+mod(n, 4)]
)$ /* R. J. Mathar, Mar 19 2012 */
(Magma) &cat[[1, 1, -1, -1]^^20]; // Vincenzo Librandi, Feb 18 2016
(PARI) a(n)=(-1)^(n\2) \\ Charles R Greathouse IV, Nov 07 2016
(Python)
def A057077(n): return -1 if n&2 else 1 # Chai Wah Wu, Jun 30 2024

Crossrefs

|a(n)|=A000012. Cf. A049310.

Cf. A000587, A121867, A121868, A130151, A143621, A143622, A143623, A143624, A143628.

Sequence in context: A332433 A376139 A374385 * A262725 A070748 A154990

Adjacent sequences: A057074 A057075 A057076 * A057078 A057079 A057080

Keyword

sign,easy

Author

Wolfdieter Lang, Aug 04 2000

Status

approved