0,1

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

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.

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

a(n)=cos(n*Pi/2)+sin(n*Pi/2) with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jun 12 2006

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]

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

A000587, A121867, A121868, A130151, A143621, A143622, A143623, A143624, A143628. [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]

Sequence in context: A076479 A155040 A033999 this_sequence A162511 A157895 A063747

Adjacent sequences: A057074 A057075 A057076 this_sequence A057078 A057079 A057080

easy,sign

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 2000