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Search: id:A140827
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| A140827 |
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Interleave denominators and numerators of convergents to sqrt(3). |
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+0
3
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| 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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Coefficients of (1+r)^m modulo r^4-r^2+1.
We have a(3*n-1)=A001075(n), a(3*n)=A001835(n-1), a(3*n+1)=A001353(n+1).
The sequence is also defined by a(0)=a(1)=1, a(k)=a(k-1)+a(k-3) when 3 divides k-1 and a(k)=a(k-1)+a(k-2) otherwise.
Satisfies a(n)^2-3*a(n-1)^2 when n=2 mod 3.
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 (ck6(AT)evansville.edu), Aug 27 2008
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REFERENCES
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Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations, arXiv:0710.2233v1 [nlin.SI].
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.
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LINKS
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Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations.
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FORMULA
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a(n) = 4*a(n-3) - a(n-6)
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EXAMPLE
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(1+r)^(2+12*q)=(-1)^q*(a(1+18*q)*(1+r^2)+a(2+18*q)*r)
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MAPLE
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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:
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CROSSREFS
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Cf. A001075, A001835, A001353, A002965.
Sequence in context: A102282 A064933 A060731 this_sequence A125621 A141001 A120415
Adjacent sequences: A140824 A140825 A140826 this_sequence A140828 A140829 A140830
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KEYWORD
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easy,nonn,uned
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AUTHOR
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Peter H. van der Kamp (peterhvanderkamp(AT)gmail.com), Jul 18 2008, Jul 22 2008
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