|
| |
|
|
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; 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 (ck6(AT)evansville.edu), 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
| Index to sequences with 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:
|
|
|
CROSSREFS
| Cf. A001075, A001835, A001353, A002965, A002530.
Sequence in context: A171027 A064933 A060731 * A125621 A141001 A196382
Adjacent sequences: A140824 A140825 A140826 * A140828 A140829 A140830
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Peter H. van der Kamp (peterhvanderkamp(AT)gmail.com), Jul 18 2008, Jul 22 2008
|
| |
|
|
