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A048788 a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2). 12
0, 1, 2, 3, 8, 11, 30, 41, 112, 153, 418, 571, 1560, 2131, 5822, 7953, 21728, 29681, 81090, 110771, 302632, 413403, 1129438, 1542841, 4215120, 5757961, 15731042, 21489003, 58709048, 80198051, 219105150, 299303201, 817711552, 1117014753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Numerators of continued fraction convergents to sqrt(3) - 1 (A160390). See A002530 for denominators. - N. J. A. Sloane, Dec 17 2007

Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ... - Clark Kimberling, Sep 21 2013

A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. - Peter Bala, Jun 06 2014

REFERENCES

Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.

LINKS

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

P. Bala, Notes on 2-periodic continued fractions and Lehmer sequences

Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.

D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1)

FORMULA

G.f.: x*(1+2*x-x^2)/(1-4*x^2+x^4). - Paul Barry, Sep 18 2009

a(n) = 4*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

a(2*n-1) = A001835(n); a(2*n) = 2*A001353(n). - Peter Bala, Jun 06 2014

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a1(n-1),a0(n)] for n>0:

a0(n) = ((3+sqrt(3))*(2-sqrt(3))^n-((-3+sqrt(3))*(2+sqrt(3))^n))/6.

a1(n) = 2*sum(i=1,n,a0(i)). (End)

a(n) = ((r + (-1)^n/r)*s^n/2^(n/2) - (1/r + (-1)^n*r)*2^(n/2)/s^n)*sqrt(6)/12, where r = 1 + sqrt(2), s = 1 + sqrt(3). - Vladimir Reshetnikov, May 11 2016

MATHEMATICA

Numerator[NestList[(2/(2 + #))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

CoefficientList[Series[x (1 + 2 x - x^2)/(1 - 4 x^2 + x^4), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 10 2013 *)

a0[n_] := ((3+Sqrt[3])*(2-Sqrt[3])^n-((-3+Sqrt[3])*(2+Sqrt[3])^n))/6 // Simplify

a1[n_] := 2*Sum[a0[i], {i, 1, n}]

Flatten[MapIndexed[{a1[#-1], a0[#]}&, Range[20]]] (* Gerry Martens, Jul 10 2015 *)

Round@Table[With[{r = 1 + Sqrt[2], s = 1 + Sqrt[3]}, ((r + (-1)^n/r) s^n/2^(n/2) - (1/r + (-1)^n r) 2^(n/2)/s^n) Sqrt[6]/12], {n, 0, 20}] (* or *) LinearRecurrence[{0, 4, 0, -1}, {0, 1, 2, 3}, 20] (* Vladimir Reshetnikov, May 11 2016 *)

PROG

(MAGMA) I:=[0, 1, 2, 3]; [n le 4 select I[n] else 4*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 10 2013

(PARI) main(size)=v=vector(size); v[1]=0; v[2]=1; v[3]=2; v[4]=3; for(i=5, size, v[i]=4*v[i-2] - v[i-4]); v; \\ Anders Hellström, Jul 11 2015

(PARI) a=vector(50); a[1]=1; a[2]=2; for(n=3, #a, if(n%2==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2])); concat(0, a) \\ Colin Barker, Jan 30 2016

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 4, 0]^n*[0; 1; 2; 3])[1, 1] \\ Charles R Greathouse IV, Mar 16 2017

CROSSREFS

Cf. A002530, A002531.

Bisections are A001835 and A052530.

Sequence in context: A289752 A119064 A285113 * A239453 A143914 A041123

Adjacent sequences:  A048785 A048786 A048787 * A048789 A048790 A048791

KEYWORD

nonn,easy,frac

AUTHOR

Robin Trew (trew(AT)hcs.harvard.edu), Dec 11 1999

EXTENSIONS

Denominator of g.f. corrected by Paul Barry, Sep 18 2009

Incorrect g.f. deleted by Colin Barker, Aug 10 2012

STATUS

approved

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Last modified June 16 06:46 EDT 2019. Contains 324145 sequences. (Running on oeis4.)