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Interleave denominators and numerators of convergents to sqrt(3).
3

%I #28 Mar 11 2020 05:17:15

%S 1,1,2,3,4,7,11,15,26,41,56,97,153,209,362,571,780,1351,2131,2911,

%T 5042,7953,10864,18817,29681,40545,70226,110771,151316,262087,413403,

%U 564719,978122,1542841,2107560,3650401,5757961,7865521,13623482,21489003,29354524,50843527,80198051,109552575

%N Interleave denominators and numerators of convergents to sqrt(3).

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

%C 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

%C From _Michel Dekking_, Mar 11 2020: (Start)

%C 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.

%C 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.

%C Any natural number N can be uniquely expanded as

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

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

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

%C (End)

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

%H Harvey P. Dale, <a href="/A140827/b140827.txt">Table of n, a(n) for n = 0..1000</a>

%H Demontigny et al., <a href="https://doi.org/10.1016/j.jnt.2014.01.028">Generalizing Zeckendorf's Theorem to f-decompositions</a>, Journal of Number Theory 141, 135-158 (2014).

%H Peter H. van der Kamp, <a href="http://arxiv.org/abs/0710.2233">Global classification of two-component approximately integrable evolution equations</a>, arXiv:0710.2233 [nlin.SI], 2007-2008.

%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997), 122-126.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4,0,0,-1).

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

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

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

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

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

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

%e Here we write N = [d(k)d(k-1)...d(0)] for the 3-bin expansion of N.

%e 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

%p 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:

%t 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 *)

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

%K easy,nonn

%O 0,3

%A _Peter H van der Kamp_, Jul 18 2008, Jul 22 2008