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A178765 a(n) = 17*a(n-1) + a(n-2), with a(-1) = 0 and a(0) = 1. 13
0, 1, 17, 290, 4947, 84389, 1439560, 24556909, 418907013, 7145976130, 121900501223, 2079454496921, 35472626948880, 605114112627881, 10322412541622857, 176086127320216450, 3003786576985302507, 51240457936070359069, 874091571490181406680, 14910797173269154272629 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

The numerators and the denominators of continued fraction convergents to (17+sqrt(293))/2 lead to the sequence given above.

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 17's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

For n>=0, a(n) equals the number of words of length n on alphabet {0,1,...,17} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..800

Dale Gerdemann, Fractal images from (17,1) recursion, YouTube Video, Nov 08 2014

Dale Gerdemann, Fractal images from (17,1) recursion: Selected image in detail, YouTube Video, Nov 08 2014

Tanya Khovanova, Recursive sequences

Index entries for linear recurrences with constant coefficients, signature (17,1).

FORMULA

a(n) = 17*a(n-1) + a(n-2) with a(-1) = 0, a(0) = 1.

G.f.: 1/(1 - 17*x - x^2).

E.g.f.: exp(17*x/2)*sinh(sqrt(293)*x/2)/(sqrt(293)/2).

a(n) = ( (17+sqrt(17^2+4))^(n+1) - (17-sqrt(17^2+4))^(n+1) )/(2^(n+1)*sqrt(17^2+4)).

a(n) = Sum_{i=0..floor(n/2)}(binomial(n+1,2*i+1)*17^(n-2*i)*(293)^i/2^n)

a(n) = Fibonacci(n+1,17), the (n+1)-th Fibonacci polynomial evaluated at x=17.

a(n) = U(n, 17*I/2)*(-I)^n with I^2=(-1) and U(n, x/2)=S(n, x), see A049310.

a(n-r-1)*a(n+r-1) - a(n-1)^2 + (-1)^(n-r)*a(r-1)^2 = 0; a(-1) = 0 and n >= r+1.

a(n-1) + a(n+1) = A090306(n+1); A090306(n)^2 - 293*a(n-1)^2 - 4*(-1)^n = 0.

a(p-1) == 293^((p-1)/2)) (mod p) for odd primes p.

a(2n+1) = 17*A098248(n) (S(n,291)), a(2n) = A098250(n) (First differences of S(n,291)).

a(3n) = A041551(5n), a(3n+1) = A041551(5n+3), a(3n+2) = 2*A041551(5n+4).

Lim_{k -> infinity}(a(n+k)/a(k)) = (A090306(n) + a(n)*sqrt(293))/2.

Lim_{n -> infinity)(A090306(n)/a(n)) = sqrt(293).

EXAMPLE

a(2) = 17*a(1) + a(0) = 289 + 1 = 290.

MAPLE

A178765:=proc(n): if n=0 then 1 elif n=1 then 17 elif n>=2 then 17*procname(n-1)+procname(n-2) fi: end: seq(A178765(n), n=0..15);

MATHEMATICA

Join[{a=0, b=1}, Table[c=17*b+1*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)

Join[{0}, LinearRecurrence[{17, 1}, {1, 17}, 30]] (* Harvey P. Dale, Jan 29 2014 *)

CoefficientList[Series[x/(1-17x-x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)

PROG

(Magma) [n le 2 select (n-1) else 17*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Nov 08 2014

(PARI) my(x='x+O('x^30)); concat([0], Vec(1/(1-17*x-x^2))) \\ G. C. Greubel, Jan 24 2019

(Sage) (x/(1-17*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 24 2019

(GAP) a:=[1, 17];; for n in [3..30] do a[n]:=17*a[n-1]+a[n-2]; od; Concatenation([0], a); # G. C. Greubel, Jan 24 2019

CROSSREFS

Cf. A000045 (k=1), A006190 (k=3), A052918 (k=5), A054413 (k=7), A099371 (k=9), A049666 (k=11), A140455 (k=13), A154597 (k=15), this sequence (k=17).

Cf. A086902, A087130.

Cf. A243399.

Sequence in context: A128358 A015969 A001026 * A041546 A186000 A222572

Adjacent sequences: A178762 A178763 A178764 * A178766 A178767 A178768

KEYWORD

nonn,easy

AUTHOR

Johannes W. Meijer, Jun 12 2010, Jul 09 2011

EXTENSIONS

Changed name from defining a(1)=17. - Jon Perry, Nov 08 2014

More terms from Vincenzo Librandi, Nov 08 2014

STATUS

approved

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Last modified December 6 15:46 EST 2022. Contains 358644 sequences. (Running on oeis4.)