|
Search: id:A057078
|
|
|
| A057078 |
|
Periodic sequence 1,0,-1...; expansion of (1+x)/(1+x+x^2). |
|
+0
21
|
|
| 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
(list; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
Partial sums of signed sequence is shifted unsigned one: |a(n+2)|= A011655(n+1).
With interpolated zeros, a(n)=sin(5*pi*n/6+pi/3)/sqrt(3)+cos(pi*n/6+pi/6)/sqrt(3); this gives the diagonal sums of the Riordan array (1-x^2, x(1-x^2)). - Paul Barry (pbarry(AT)wit.ie), Feb 02 2005
|
|
LINKS
|
Ralph E. Griswold, Shaft Sequences
Index entries for sequences related to Chebyshev polynomials.
|
|
FORMULA
|
a(n)=S(n, -1)+S(n-1, -1) = S(2*n, 1); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, -1)= A049347(n). S(n, 1)= A010892(n).
G.f.: (1+x)/(1+x+x^2).
a(n)=(1/2)((-1)^floor(2n/3)+(-1)^floor((2n+1)/3)). a(n)=-a(n-1)-a(n-2). a(n)=A061347(n)-A049347(n+2). - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
a(n)=sum C(n+k, 2k)(-1)^(n-k), k=0, .., n = sum C(n+1-k, k)(-1)^(n-k), k=0, .., floor((n+1)/2). - Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2003
Binomial transform is A010892. a(n)=2sqrt(3)sin(2pi*n/3+pi/3)/3 - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003
a(n)=cos(2*pi*n/3)+sin(2*pi*n/3)/sqrt(3). - Paul Barry (pbarry(AT)wit.ie), Oct 27 2004
a(n)=sum{k=0..n, (-1)^A010060(2n-2k)*mod(binomial(2n-k, k), 2)} - Paul Barry (pbarry(AT)wit.ie), Dec 11 2004
a(n) = -(1/3)*[2*(n mod 3)-(n+1) mod 3-(n+2) mod 3] - Paolo P. Lava (ppl(AT)spl.at), Oct 09 2006
a(n)=(4/3)*(|sin(pi*(n-2)/3)|-|sin(pi*n/3)|)*|sin(pi*(n-1)/3)|. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n)=1-(n mod 3)=1+3*floor(n/3))-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
a(n)=1-A010872(n)=1+3*A002264(n)-n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
Euler transform of length 3 sequence [ 0, -1, 1]. - Michael Somos Oct 15 2008
|
|
EXAMPLE
|
1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ...
|
|
PROGRAM
|
(PARI) {a(n) = [1, 0, -1][n%3 + 1]} /* Michael Somos Oct 15 2008 */
|
|
CROSSREFS
|
A049310, A010892, A011655.
A049347(n) = a(-n).
Sequence in context: A071036 A166946 A141687 this_sequence A127245 A088150 A117567
Adjacent sequences: A057075 A057076 A057077 this_sequence A057079 A057080 A057081
|
|
KEYWORD
|
easy,sign
|
|
AUTHOR
|
Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 2000
|
|
|
Search completed in 0.002 seconds
|
