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A079935 a(n) = 4*a(n-1) - a(n-2). 21
1, 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, 413403, 1542841, 5757961, 21489003, 80198051, 299303201, 1117014753, 4168755811, 15558008491, 58063278153, 216695104121, 808717138331, 3018173449203, 11263976658481 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A001835 for another version.

Greedy frac multiples of sqrt(3): a(1)=1, sum(n>0,frac(a(n)*x)) < 1 at x=sqrt(3).

The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1). - Paul Barry, Mar 13 2005

The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007

From Richard Choulet, May 09 2010: (Start)

This sequence is a particular case of the following situation:

a(0)=1, a(1)=a, a(2)=b with the recurrence relation a(n+3) = (a(n+2)*a(n+1)+q)/a(n)

where q is given in Z to have Q=(a*b^2+q*b+a+q)/(a*b) itself in Z.

The g.f is f: f(z)=(1+a*z+(b-Q)*z^2+(a*b+q-a*Q)*z^3)/(1-Q*z^2+z^4);

so we have the linear recurrence: a(n+4)=Q*a(n+2)-a(n).

The general form of a(n) is given by:

a(2*m) = sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(b-Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)) and

a(2*m+1) = a*sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(a*b+q-a*Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)).

(End)

x-values in the solution to 3*x^2-2=y^2. - Sture Sjöstedt, Nov 25 2011

LINKS

Table of n, a(n) for n=1..24.

Tanya Khovanova, Recursive Sequences

Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.

Index entries for linear recurrences with constant coefficients, signature (4,-1).

FORMULA

For n>0, a(n)= ceil( (2+sqrt(3))^n/(3+sqrt(3)) ).

G.f.: (1-x)/(1-4x+x^2); E.g.f.: exp(2x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); a(n)=(1/2+sqrt(3)/6)(2+sqrt(3))^n+(1/2-sqrt(3)/6)(2-sqrt(3))^n (offset 0). Binomial transform of A002605. - Paul Barry, Sep 17 2003

a(n) = sum{k=0..n, binomial(2n-k, k)2^(n-k)}. - Paul Barry, Jan 22 2005

a(n) = (-1)^n*U(2n, I*sqrt(2)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry, Mar 13 2005

a(n) = Jacobi_P(n,-1/2,1/2,2)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006

a(n) = sqrt(2+(2-sqrt(3))^(2*n-1)+(2+sqrt(3))^(2*n-1))/sqrt(6). - Gerry Martens, Jun 05 2015

a(n) = (1/2+sqrt(3)/6)*(2-sqrt(3))^n + (1/2 - sqrt(3)/6)*(2+sqrt(3))^n. - Robert Israel, Jun 05 2015

EXAMPLE

a(4) = 41 since frac(1*x) + frac(3*x) + frac(11*x) + frac(41*x) < 1, while frac(1*x) + frac(3*x) + frac(11*x) + frac(k*x) > 1 for all k>11 and k<41.

MAPLE

f:= gfun:-rectoproc({a(n) = 4*a(n-1) - a(n-2), a(1)=1, a(2)=3}, a(n), remember):

seq(f(n), n=1..30); # Robert Israel, Jun 05 2015

MATHEMATICA

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}]] (* Robert G. Wilson v, Jan 13 2005 *)

LinearRecurrence[{4, -1}, {1, 3}, 30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^2), {x, 0, 30}], x]  (* Harvey P. Dale, Apr 26 2011 *)

PROG

(Sage) [lucas_number1(n, 4, 1)-lucas_number1(n-1, 4, 1) for n in xrange(1, 25)] # Zerinvary Lajos, Apr 29 2009

(Haskell)

a079935 n = a079935_list !! (n-1)

a079935_list =

   1 : 3 : zipWith (-) (map (4 *) $ tail a079935_list) a079935_list

-- Reinhard Zumkeller, Aug 14 2011

(MAGMA) I:=[1, 3]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 06 2015

(PARI) a(n)=([0, 1; -1, 4]^(n-1)*[1; 3])[1, 1] \\ Charles R Greathouse IV, Mar 18 2017

CROSSREFS

Cf. A002530 (denominators of convergents to sqrt(3)), A079934, A079936, A001353.

Cf. A001835 (same except for the first term).

Row 4 of array A094954.

Cf. similar sequences listed in A238379.

Sequence in context: A077831 A032952 A001835 * A281593 A113437 A076540

Adjacent sequences:  A079932 A079933 A079934 * A079936 A079937 A079938

KEYWORD

nonn,easy,changed

AUTHOR

Benoit Cloitre and Paul D. Hanna, Jan 20 2003

STATUS

approved

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Last modified March 24 06:12 EDT 2017. Contains 283984 sequences.