login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005246 a(n) = (1 + a(n-1)*a(n-2))/a(n-3), a(0) = a(1) = a(2) = 1.
(Formerly M0829)
32
1, 1, 1, 2, 3, 7, 11, 26, 41, 97, 153, 362, 571, 1351, 2131, 5042, 7953, 18817, 29681, 70226, 110771, 262087, 413403, 978122, 1542841, 3650401, 5757961, 13623482, 21489003, 50843527, 80198051, 189750626, 299303201, 708158977, 1117014753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

For n >= 4 we have the linear recurrence a(n) = 4*a(n-2) - a(n-4). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 04 2001

Integer solutions to the equation floor(sqrt(3)*x^2)=x*floor(sqrt(3)*x). - Benoit Cloitre, Mar 18 2004

For n>2, a(n) is the smallest integer > a(n-1) such that sqrt(3)*a(n) is closer to and greater than an integer than sqrt(3)*a(n-1). i.e. a(n) is the smallest integer > a(n-1) such that frac(sqrt(3)*a(n))<frac(sqrt(3)*a(n-1)). - Benoit Cloitre, Jan 20 2003

The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; essentially, numerators=A143643 and denominators=A005246. - Clark Kimberling, Aug 27 2008

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)). [Richard Choulet, Feb 24 2010]

In the closed form formula (sqrt(2+sqrt(3))^n)=((sqrt(6)+sqrt(2))/2)^n;(-sqrt(2+sqrt(3))^n)=((-sqrt(6)-sqrt(2))/2)^n;(sqrt(2-sqrt(3))^n)=((sqrt(6)-sqrt(2))/2)^n ;(-sqrt(2-sqrt(3))^n)=((sqrt(2)-sqrt(6))/2)^n. - Tim Monahan, Jul 07 2011

a(n) for n>1 are the integer square roots of (floor(m^2/3)+1), where the values of m are given by A143643.  Also see A082630. - Richard R. Forberg, Nov 14 2013

The a(n) = (1 + a(n-1)*a(n-2))/a(n-3) recursion has the Laurent property. If a(0), a(1), a(2) are variables, then a(n) is a Laurent polynomial (a rational function with a monomial denominator). - Michael Somos, Feb 27 2019

REFERENCES

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

Reid Barton, A combinatorial interpretation of the sequence 1, 1, 2, 6, 21, 77, ...

Reid Barton, A combinatorial interpretation of the sequence 1, 1, 2, 6, 21, 77, ..., [Annotated scanned copy]

Peter Cameron's Blog, The ADE affair, 3, Posted 23/06/2011.

T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.

Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003.

R. K. Guy, Letter to N. J. A. Sloane, Feb 1986

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Valentin Ovsienko, Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 10.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

FORMULA

G.f.: (1+x-3*x^2-2*x^3)/(1-4*x^2+x^4).

lim n ->infinity a(2n+1)/a(2n) = (3+sqrt(3))/3 =1.5773502... lim n ->infinity a(2n)/a(2n-1) = (3+sqrt(3))/2 = 2.3660254.... - Benoit Cloitre, Aug 07 2002

A101265(n) = a(n)*a(n+1). - Franklin T. Adams-Watters, Apr 24 2006

a(n) = a(2 - n) for all n in Z. - Michael Somos, Nov 15 2006

a(2*n + 1) = A001075(n). a(2*n) = A001835(n). a(2*n + 1) - a(2*n) = a(2*n + 2) - a(2*n + 1) = A001353(n). - Michael Somos, May 24 2012

For n>2: a(n) = a(n-1) + SUM(a(2*k): 1 <= k < n/2). - Reinhard Zumkeller, Dec 16 2007

a(2*m) = sum((-1)^p*binomial(m-p,p)*4^(m-2*p),p=0..floor(m/2))-3*sum((-1)^p*binomial(m-1-p,p)*4^(m-1-2*p),p=0..floor((m-1)/2)). a(2*m+1)=sum((-1)^p*binomial(m-p,p)*4^(m-2*p),p=0..floor(m/2))-2*sum((-1)^p*binomial(m-1-p,p)*4^(m-1-2*p),p=0..floor((m-1)/2)). [Richard Choulet, Feb 24 2010]

From Tim Monahan, Jul 01 2011:  (Start)

Closed form without extra leading 1 ((sqrt(6)+3)*(sqrt(2+sqrt(3))^n+(sqrt(2-sqrt(3))^n))+(3-sqrt(6))*(-sqrt(2+sqrt(3))^n+(-sqrt(2-sqrt(3))^n)))/12.

Closed form with extra leading 1 ((6+3*sqrt(6)-2*sqrt(3)-3*sqrt(2))*(sqrt(2+sqrt(3))^n)+(6+3*sqrt(6)+2*sqrt(3)+3*sqrt(2))*(sqrt(2-sqrt(3))^n)+(6-3*sqrt(6)-2*sqrt(3)+3*sqrt(2))*(-sqrt(2+sqrt(3))^n)+(6-3*sqrt(6)+2*sqrt(3)-3*sqrt(2))*(-sqrt(2-sqrt(3))^n))/24.  (End)

a(2*n+2) = sum {k = 0..n} 2^k*binomial(n+k,2*k); a(2*n+1) = sum {k = 0..n} n/(n+k)*2^k*binomial(n+k,2*k) for n >= 1. Row sums of A211956. - Peter Bala, May 01 2012

a(n) = ((sqrt(2)+sqrt(3)+(-1)^n*(sqrt(2)-sqrt(3)))*sqrt(2+(2-sqrt(3))^n*(2+ sqrt(3))-(-2+sqrt(3))*(2+ sqrt(3))^n))/(4*sqrt(3)). - Gerry Martens, Jun 06 2015

0 = a(n) - 2*a(n+1) + a(n+2) if n is even, 0 = a(n) - 3*a(n+1) + a(n+2) if n is odd for all n in Z. - Michael Somos, Feb 10 2017

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 11*x^6 + 26*x^7 + 41*x^8 + ...

a(4)=4^2-4^0-3*4^1=3. a(7)=4^3-4*binomial(2,1)-2*(4^2-1)=26. [Richard Choulet, Feb 24 2010]

MAPLE

A005246:=-(-1-z+2*z**2+z**3)/(1-4*z**2+z**4); # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for one of the leading 1's.

for q from 1 to 10 do :a:=1:b:=1:Q:=(a*b^2+q*b+a+q)/(a*b): for m from 0 to 15 do U(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)):od: for m from 0 to 15 do V(m):=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)):od:for m from 0 to 15 do W(2*m):=U(m):od:for m from 0 to 14 do W(2*m+1):=V(m):od:seq(W(m), m=0..30):od; # Richard Choulet, Feb 24 2010

MATHEMATICA

RecurrenceTable[{a[0]==a[1]==a[2]==1, a[n]==(1+a[n-1]a[n-2])/a[n-3]}, a, {n, 40}] (* Harvey P. Dale, May 28 2013 *)

a[n_] := Cosh[(n-1)*ArcSinh[1/Sqrt[2]]]*If[EvenQ[n], Sqrt[2/3], 1]; Table[a[n] // FunctionExpand, {n, 0, 34}] (* Jean-François Alcover, Dec 10 2014, after Peter Bala *)

a[ n_] := With[{m = If[ n < 0, 2 - n, n]}, SeriesCoefficient[ (1 + x - 3 x^2 - 2 x^3) / (1 - 4 x^2 + x^4), {x, 0, m}]]; (* Michael Somos, Feb 10 2017 *)

PROG

(PARI) {a(n) = if( n<0, n = 2 - n); polcoeff((1 + x - 3*x^2 - 2*x^3) / (1 - 4*x^2 + x^4) + x * O(x^n), n)}; /* Michael Somos, Nov 15 2006 */

(PARI) {a(n) = real( (2 + quadgen(12))^(n\2) * if( n%2, 1, 1 - 1 / quadgen(12)) )}; /* Michael Somos, May 24 2012 */

(Haskell)

a005246 n = a005246_list !! n

a005246_list = 1 : 1 : 1 : map (+ 1) (zipWith div

   (zipWith (*) (drop 2 a005246_list) (tail a005246_list)) a005246_list)

-- Reinhard Zumkeller, Mar 07 2012

CROSSREFS

Bisections are A001835 and A001075.

Cf. A101265. Row sums of A211956.

Cf. A001353.

Sequence in context: A121268 A101173 A294451 * A116406 A112843 A036651

Adjacent sequences:  A005243 A005244 A005245 * A005247 A005248 A005249

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Michael Somos, Aug 01 2001

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 31 03:48 EDT 2020. Contains 333136 sequences. (Running on oeis4.)