A140827 Interleave denominators and numerators of convergents to sqrt(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

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

From Michel Dekking, Mar 11 2020: (Start)

This sequence can be seen as a generalization of the Fibonacci numbers A000045. The Zeckendorf expansion of a natural number uses the Fibonacci numbers as constituents. The Zeckendorf expansion is called a 2-bin decomposition in the paper by Demontigny et al.

The numbers a(n) are the constituents of the 3-bin decomposition of a natural number. See Example 4.2 and Proposition 4.3 in the Demontigny et al. paper.

Any natural number N can be uniquely expanded as

N = Sum_{i=0..k} d(i)*a(i)

under the requirement d(i)d(i+1) = 0, and d(3i)d(3i+2) = 0 for all i.

Here k is the largest integer such that a(k) < N+1.

(End)

References

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

Links

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

Demontigny et al., Generalizing Zeckendorf's Theorem to f-decompositions, Journal of Number Theory 141, 135-158 (2014).

Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations, arXiv:0710.2233 [nlin.SI], 2007-2008.

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

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

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).
Here we write N = [d(k)d(k-1)...d(0)] for the 3-bin expansion of N.
0=[0], 1 =[1], 2=[10], 3=[100], 4=[1000], 5=[1001], 6=[1010], 7=[10000], 8=[10001], 9=[10010], 10=[10100], 11=[100000]. - Michel Dekking, Mar 11 2020

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 A333260

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