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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A140827 Interleave denominators and numerators of convergents to sqrt(3). 3
1, 1, 2, 3, 4, 7, 11, 15, 26, 41, 56, 97, 153, 209, 362, 571, 780, 1351, 2131, 2911, 5042, 7953, 10864, 18817, 29681, 40545, 70226, 110771, 151316, 262087, 413403, 564719, 978122, 1542841, 2107560, 3650401, 5757961, 7865521, 13623482, 21489003, 29354524, 50843527, 80198051, 109552575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Coefficients of (1+r)^m modulo r^4-r^2+1.

The first few principal and intermediate convergents to 3^(1/2) are 1/1, 2/1, 3/2, 5/3, 7/4, 12/7; essentially, numerators=A143642 and denominators=A140827. - Clark Kimberling, Aug 27 2008

REFERENCES

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

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

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations, arXiv:0710.2233 [nlin.SI].

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

FORMULA

a(n) = 4*a(n-3) - a(n-6).

G.f.: ( 1+x+2*x^2-x^3-x^5 ) / ( 1-4*x^3+x^6 ).

a(n) = a(n-1)+a(n-3) if 3 |(n-1), else a(n)=a(n-1)+a(n-2), with n>1.

a(3*n-1)=A001075(n); a(3*n)=A001835(n-1); a(3*n+1)=A001353(n+1).

a(n)^2 = 1+3*a(n-1)^2 if n==2 (mod 3).

EXAMPLE

(1+r)^(2+12*q)=(-1)^q*(a(1+18*q)*(1+r^2)+a(2+18*q)*r)

MAPLE

N:=100: a[0]:=1: a[1]:=1: for i from 2 to N do if i mod 3 = 1 then a[i]:=a[i-1]+a[i-3] else a[i]:=a[i-1]+a[i-2] fi od:

MATHEMATICA

idnc[n_]:=Module[{cvrgts=Convergents[Sqrt[3], n], num, den}, num=Take[ Numerator[ cvrgts], {2, -1, 2}]; den=Denominator[cvrgts]; Riffle[den, num, 3]]; idnc[30] (* Harvey P. Dale, Mar 17 2012 *)

CROSSREFS

Cf. A001075, A001835, A001353, A002965, A002530.

Sequence in context: A064933 A060731 A238492 * A125621 A141001 A196382

Adjacent sequences:  A140824 A140825 A140826 * A140828 A140829 A140830

KEYWORD

easy,nonn

AUTHOR

Peter H van der Kamp, Jul 18 2008, Jul 22 2008

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 April 26 04:04 EDT 2019. Contains 322469 sequences. (Running on oeis4.)