login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054413 a(n) = 7*a(n-1) + a(n-2). 32
1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, 47301793, 337737401, 2411463600, 17217982601, 122937341807, 877779375250, 6267392968557, 44749530155149, 319514104054600, 2281348258537349, 16288951913816043, 116304011655249650, 830417033500563593 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In general, sequences with recurrence a(n)= k*a(n-1)+a(n-2) and a(0)=1 [and a(-1)=0] have generating function 1/(1-k*x-x^2). If k is odd (k>=3) 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. - Gary W. Adamson, Feb 22 2009

Contribution from Johannes W. Meijer, Jun 12 2010: (Start)

For the sequence given above k=7 which implies that it is associated with A041091.

For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0) = 2 [and a(-1) = 0] see A086902; a sequence that is associated with A041090.

For more information follow the Khovanova link and see A087130, A140455 and A178765.

(End)

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 7's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

a(n) equals the number of words of length n on alphabet {0,1,...,7} avoiding runs of zeroes of odd lengths. - Milan Janjic, Jan 28 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Sergio Falcon and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.

Sergio Falcon and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (7,1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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, 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, 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, Jun 25 2008

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

Contribution from Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 7*A097836(n), a(2n) = A097838(n).

Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.

Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53).

(End)

Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = (sqrt(53)-7)/2. - Vladimir Shevelev, Feb 23 2013

MATHEMATICA

a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*7, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

LinearRecurrence[{7, 1}, {1, 7}, 30] (* Vincenzo Librandi, Feb 23 2013 *)

PROG

(Sage) [lucas_number1(n, 7, -1) for n in xrange(1, 19)] # Zerinvary Lajos, Apr 24 2009

(MAGMA) I:=[1, 7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013

(PARI) a(n)=([0, 1; 1, 7]^n*[1; 7])[1, 1] \\ Charles R Greathouse IV, Apr 08 2016

CROSSREFS

Cf. A099367, A000045, A000129, A006190, A001076, A052918, A005668, A243399.

Sequence in context: A096882 A033125 A022037 * A163458 A081571 A275827

Adjacent sequences:  A054410 A054411 A054412 * A054414 A054415 A054416

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, May 10 2000

EXTENSIONS

Formula corrected by Johannes W. Meijer, May 30 2010, Jun 02 2010

Extended by T. D. Noe, May 23 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 26 05:17 EDT 2017. Contains 284111 sequences.