

A275974


Partial sums of the Jeffreys binary sequence A275973.


2



1, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 43, 43, 43, 43
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OFFSET

1,3


COMMENTS

The ratio d(n) = a(n)/n, while obviously bounded, has no limit. Rather, it kind of 'oscillates', at an exponentially decreasing rate, between about 1/3 and 2/3. As mentioned by Jeffreys, the values of liminf and limsup of the set {d(n)} are 1/3 and 2/3, respectively. A proof of this fact by elementary means is relatively easy, for example using the first formula below, but the following statement is a conjecture: Any real value c comprised in the interval [1/3,2/3] is an accumulation point of {d(n)}.


REFERENCES

HsienKuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wpcontent/files/2016/12/aathhrr1.pdf. Also Exact and Asymptotic Solutions of a DivideandConquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585


LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..2100


FORMULA

For k>=0 and 4^k <= m <= 2*4^k (i.e., m spanning a block of 0's in A275974), a(m) = 1+2*(1+4+4^2+...+4^(k1)) = (1/3)+(2/3)*4^k.
For any n, d(n) = a(n)/n > 1/3.
liminf_{n>infinity}d(n) = 1/3 and limsup_{n>infinity}d(n) = 2/3.


PROG

(PARI) JeffreysSequence(nmax) = {
my(a=vector(nmax), n=0, p=1); a[n++]=1;
while(n<nmax,
for(k=2^(p1)+1, 2^p, a[n++]=0; if(n==nmax, break));
if(n<nmax, for(k=2^p+1, 2^(p+1), a[n++]=1; if(n==nmax, break)));
p+=2; );
return(a); }
a = JeffreysSequence(2100);
for(n=2, #a, a[n]+=a[n1]); a


CROSSREFS

Cf. A275973.
Sequence in context: A120506 A277267 A246262 * A052288 A280455 A284725
Adjacent sequences: A275971 A275972 A275973 * A275975 A275976 A275977


KEYWORD

nonn


AUTHOR

Stanislav Sykora, Aug 15 2016


STATUS

approved



