|
| |
|
|
A006340
|
|
An "eta-sequence": [ (n+1)*tau + 1/2 ] - [ n*tau + 1/2 ], tau = (1 + sqrt(5))/2.
(Formerly M0100)
|
|
5
|
|
|
|
2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Equals its own "derivative", which is formed by counting the strings of 1's that lie between 2's.
Conjecture: A006340 = continued fraction expansion of (2.729967741... = sup{f(n,1)}), where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in the lower Wythoff sequence (A000201), else f(n,x) = 1/x. The first 12 values of f(n,1) are given in Example at A245216. - Clark Kimberling, Jul 14 2014
The description of this sequence is not correct, since the derivative of a equals
a' = 1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,...
The claim by Hofstadter in formula (4) in the 1977 letter to Sloane is also not correct, since the second derivative of a is equal to
a'' = 2,2,1,2,1,2,2,1,2,1,2,2,1,...
so a is not equal to its own second derivative.
Nevertheless, this sequence has a self-similarity property: if one replaces every chunk 212 with 1 and every chunk 21212 with 2, then one obtains back the original sequence. In other words, (a(n)) is the unique fixed point of the morphism sigma given by sigma: 1->212, 2->21212.
This can be proved following the ideas of Chapter 2 in Lothaire's book and Section 4 of my paper "Substitution invariant Sturmian words and binary trees".
To comply with these references change the alphabet to {0,1}. This changes sigma into the morphism 0->101, 1->10101.
The fractional part {tau} of tau is larger than 1/2; as it is convenient to have it smaller than 1/2 we change to beta = 1-tau = (3-sqrt(5))/2.
This changes the morphism 0->101, 1->10101 to its mirror image psi given by 0->01010, 1->010.
Let psi_1 and psi_2 be the elementary Sturmian morphisms given by
psi_1(0)=01 , psi_1(1)=1, psi_2(0)=10, psi_2(1)=0.
Then psi = psi_2^2 psi_1.
This already shows that psi generates a Sturmian sequence with certain parameters alpha and rho: s(alpha,rho) = ([(n+1)*alpha+rho]-[n*alpha+rho]).
Since psi is the composition psi_2^2psi_1, the parameters of s(alpha,rho) are given by the composition T:=T_2^2T_1 of the fractional linear maps
T_1(x,y) = ((1-x)/(2-x),(1-y)/(2-x)),
T_2(x,y) = ((1-x)/(2-x), (2-x-y)/(2-x)).
Since one can verify that T(beta,1/2)=(beta,1/2), it follows that
alpha = beta, and rho = 1/2.
(End)
|
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) rt(n) = my(tau=(1 + sqrt(5))/2); round(tau*n)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
D. R. Hofstadter, Jul 15 1977
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
