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
Links
- 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.
- StackExchange, Finding sum(k=1..oo) (-1)^(k(k+1)/2), (2025) provides Dirichlet sums.
- Index entries for linear recurrences with constant coefficients, signature (0,-1).
- Index entries for sequences related to Chebyshev polynomials.
Crossrefs
Programs
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Magma
&cat[[1, 1, -1, -1]^^20]; // Vincenzo Librandi, Feb 18 2016
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Maple
seq((-1)^floor(k/2),k=0..70); # Wesley Ivan Hurt, Jul 06 2013 A057077 := proc(n) op(1+(n mod 4),[1,1,-1,-1]) ; end proc: seq(A057077(n),n=0..40) ; # R. J. Mathar, Feb 27 2025
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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 *) -
Maxima
A057077(n) := block( [1,1,-1,-1][1+mod(n,4)] )$ /* R. J. Mathar, Mar 19 2012 */
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PARI
a(n)=(-1)^(n\2) \\ Charles R Greathouse IV, Nov 07 2016
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Python
def A057077(n): return -1 if n&2 else 1 # Chai Wah Wu, Jun 30 2024
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
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
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
Comments