0,1

Sequence related to the 'silver mean' [5;5,5,5,5,...].

The solution to the general recurrence a(n)=(2k+1)a(n-1)+a(n-2), a(0)=2, a(1)=2k+1 is a(n)=((2k+1)+sqrt(4k^2+4k+5))^n+(2k+1)-sqrt(4k^2+4k+5))^n)/2; a(n)=2^(1-n)sum{j=0..n, C(n, 2j)(4k^2+4k+5)^j(2k+1)^(n-2j)}; a(n)=2T(n, (2k+1)x/2)(-1)^i with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

Primes in this sequence include a(0) = 2; a(1) = 5; a(4) = 727; a(8) = 528527 (3) semiprimes in this sequence include a(7) = 101785; a(13) = 1995189565; a(16) = 279340789727; a(19) = 39109705751345; a(20) = 203080369893127 - Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 09 2005

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

a(n)=((5+sqrt(29))/2)^n+((5-sqrt(29))/2)^n

E.g.f. : 2exp(5x/2)cosh(sqrt(29)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)29^k5^(n-2k)}. a(n)=2T(n, 5i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

O.g.f.: (-2+5*x)/(-1+5*x+x^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 02 2007

Cf. A006497, A014448, A085447.

Cf. A086902, A000032.

Equals A100236(n) + 1.

Sequence in context: A041571 A042259 A100105 this_sequence A097565 A079716 A058182

Adjacent sequences: A087127 A087128 A087129 this_sequence A087131 A087132 A087133

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Aug 16 2003