0,2

In general sequences with recurrence a(n)=(2k+1)*a(n-1)+a(n-2) and a(0)=1 [and a(-1)=0] have generating function 1/(1-kx-x^2). If k is odd they satisfy a(3n)=b(5n), a(3n+1)=b(5n+3), a(3n+2)=2*b(5n+4) where b(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2/4+1).]

a(p) == 53^((p-1)/2)) mod p, for odd primes p. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009]

S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (2007)

S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

a(3n)=A041091(5n), a(3n+1)=A041091(5n+3), a(3n+2)=2*A041091(5n+4) G.f.: 1/(1-7x-x^2)

a(n)= U(n, 7*I/2)*(-I)^n with I^2=-1 and Chebyshev's U(n, x/2)=S(n, x) polynomials. See A049310.

a(n)=F(n, 7), the n-th Fibonacci polynomial evaluated at x=7. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006

a(n) = (sigma^n-(-sigma)^{-n})/(Sqrt[53]) with sigma=(7+Sqrt[53])/2; a(n) = Sum_0^{Floor[(n-1)/2]} Binomial[n-1-i,i]*7^{n-1-2i} - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007

a(n)=-(7/106)*sqrt(53)*[7/2-(1/2)*sqrt(53)]^n+(1/2)*[7/2+(1/2)*sqrt(53)]^n+(1/2)*[7/2-(1/2) *sqrt(53)]^n+(7/106)*[7/2+(1/2)*sqrt(53)]^n*sqrt(53), with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jun 25 2008

a(n)=((7+sqrt53)^n-(7-sqrt53)^n)/(2^n*sqrt53). Offset 1. a(3)=50. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 17 2009]

a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*7, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]

(Other) sage: [lucas_number1(n, 7, -1) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]

Cf. A000045, A000129, A006190, A001076, A052918, A005668.

Cf. A099367 (squares).

Cf. A000045, A000129, A006190, A001076, A052918, A005668.

Sequence in context: A096882 A033125 A022037 this_sequence A163458 A081571 A081189

Adjacent sequences: A054410 A054411 A054412 this_sequence A054414 A054415 A054416

nonn

Henry Bottomley (se16(AT)btinternet.com), May 10 2000