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A206916 Index of the least binary palindrome >=n; also the "upper inverse" of A006995. 4
1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The least m such that A006995(m)>=n;

n is palindromic iff a(n)=A206915(n);

a(n) is the number of the binary palindrome A206914(n);

if n is a binary palindrome, then A006995(a(n))=n, so a(n) is 'inverse' with respect to A006995

LINKS

Table of n, a(n) for n=0..70.

Cf. A006995, A206920.

FORMULA

a(n)=min(m|A006995(m)>=n);

a(A006995(n))=n;

A006995(a(n))>=n, equality holds true iff n is a binary palindrome;

Let p=A206914(n), m=floor(log_2(p)) and p>2, then:

a(n)=(((5-(-1)^m)/2) + sum_{k=1..floor(m/2)} (floor(p/2^k) mod 2)/2^k))*2^floor(m/2);

a(n)=(1/2)*((6-(-1)^m)*2^floor(m/2)-1-sum_       {k=1..floor(m/2)} (-1)^floor(p/2^k)*2^(floor(m/2)-k)));

a(n)=(5-(-1)^m)*2^floor(m/2)/2-3*sum_{k=2..floor(m/2)} floor(p/2^k)*2^floor(m/2)/2^k)+(floor(p/2)*2^floor(m/2)/2-2*floor((p/2)*2^floor(m/2))*floor((m-1)/m+1/2).

Partial sums S(n) = sum_{k=0..n} a(k):

S(n) = 1+n*a(n)-A206920(a(n)-1), valid for n>0.

G.f.: g(x)=(x+x^2+x^3+sum_{j=1..infinity} x^(3*2^j)*(f_j(x)+f_j(1/x)))/(x(1-x)), where the f_j(x) are defined as follows:

  f_1(x)=x, and for j>1,

  f_j(x)=x^3*product_{k=1..floor((j-1)/2)} (1+x^b(j,k)), where b(j,k)=2^(floor((j-1)/2)-k)*((3+(-1)^j)*2^(2*k+1)+4) for k>1, and b(j,1)=(2+(-1)^j)*2^(floor((j-1)/2)+1).

EXAMPLE

a(2)=3 since 3 is the index number of the least binary palindrome >= 2;

a(5)=4 since 4 is the index number of the least binary palindrome >= 5;

a(10)=7 since A006995(7)=15>=10, but A006995(6)=9<10, and so that, 7 is the index number of least binary palindrome >= 10;

CROSSREFS

Sequence in context: A322921 A030581 A113609 * A067086 A254531 A005410

Adjacent sequences:  A206913 A206914 A206915 * A206917 A206918 A206919

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Feb 17 2012

STATUS

approved

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)