0,3

a(p) == 85^((p-1)/2)) mod p for odd primes p [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009]

J. H. Han and M. D. Hirschhorn, "Another Look at an Amazing Identity of Ramanujan", Mathematics Magazine, Vol. 79 (2006), pp. 302-304. See equation 6 on page 303.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1 - 9*x - x^2).

a(n) = 9*a(n-1)+a(n-2), n>=2, a(0)=0, a(1)=1.

a(n) = ((-i)^(n-1))*S(n-1, 9*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.

a(n)= (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2=-1/ap (Binet form).

a(n)= sum(binomial(n-1-k, k)*9^(n-1-2*k), k=0..floor((n-1)/2)), n>=1.

a(n)=F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006

a(n)=((9+sqrt85)^n-(9-sqrt85)^n)/(2^n*sqrt85). Offset 1. a(3)=82. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009]

a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*9, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]

sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(0, 1, 9, 9, 1, 0) sage: [it.next() for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008

(Other) sage: [lucas_number1(n, 9, -1) for n in xrange(0, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]

Cf. A099372 (squares).

Cf. A099371.

Sequence in context: A067506 A033119 A033127 this_sequence A068109 A163460 A081191

Adjacent sequences: A099368 A099369 A099370 this_sequence A099372 A099373 A099374

nonn,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004