Search: id:A057077
Results 1-1 of 1 results found.

%I A057077
%S A057077 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,
%T A057077 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,
%U A057077 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%V A057077 1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,
%W A057077 1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,
%X A057077 1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1
%N A057077 Periodic sequence 1,1,-1,-1...; expansion of (1+x)/(1+x^2).
%H A057077 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057077 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057077 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.
%F A057077 G.f.: (1+x)/(1+x^2).
%F A057077 a(n)=cos(n*Pi/2)+sin(n*Pi/2) with n>=0 - Paolo P. Lava (ppl(AT)spl.at), 
               Jun 12 2006
%F A057077 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..inf} (-1)^floor(n/r)*n^k/n!, k = 0,1,2,... 
               . For particular cases see A000587 (r = 1) and A143628 (r = 3). [From 
               Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]
%Y A057077 |a(n)|=A000012. Cf. A049310.
%Y A057077 A000587, A121867, A121868, A130151, A143621, A143622, A143623, A143624, 
               A143628. [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]
%Y A057077 Sequence in context: A076479 A155040 A033999 this_sequence A162511 A157895 
               A063747
%Y A057077 Adjacent sequences: A057074 A057075 A057076 this_sequence A057078 A057079 
               A057080
%K A057077 easy,sign
%O A057077 0,1
%A A057077 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 
               2000

Search completed in 0.002 seconds