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A206915 The index (in A006995) of the greatest binary palindrome <= n; also the 'lower inverse' of A006995. 13
1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The greatest m such that A006995(m)<= n;

The number of binary palindromes <= n;

n is palindromic iff a(n)=A206916(n);

a(n) is the number of the binary palindrome A206913(n);

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

Partial sums of the binary palindromic characteristic function A178225.

LINKS

Table of n, a(n) for n=0..71.

FORMULA

a(n) = max(m | A006995(m) <= n);

a(A006995(n)) = n;

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

Let p = A206913(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) = (n+1)*a(n) - A206920(a(n)).

G.f.: g(x) = (1+x+x^3+sum_{j>=1} x^(3*2^j)*(f_j(x)+f_j(1/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(1)=2 since 2 is the index number of the greatest binary palindrome <= 1;

a(5)=4 since there are only 4 binary palindromes (namely 0,1,3 and 5) which are less than or equal to 5;

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

CROSSREFS

Cf. A006995, A206914, A206915, A206916, A206920.

Sequence in context: A194964 A029923 A096029 * A210609 A210608 A095773

Adjacent sequences:  A206912 A206913 A206914 * A206916 A206917 A206918

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Feb 15 2012

STATUS

approved

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Last modified April 20 02:06 EDT 2019. Contains 322291 sequences. (Running on oeis4.)