A206913 - OEIS

A206913

Greatest binary palindrome <= n; the binary palindrome floor function.

0, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 9, 9, 9, 9, 15, 15, 17, 17, 17, 17, 21, 21, 21, 21, 21, 21, 27, 27, 27, 27, 31, 31, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 45, 45, 45, 45, 45, 45, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 63, 63, 65, 65, 65, 65

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Also the greatest binary palindrome < n + 1;

For n > 0, a(n-1) is the greatest binary palindrome < n.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

FORMULA

Let n > 2, p = 1 + 2*floor((n-1)/2), m = floor(log_2(p)), q = floor((m+1)/2), s = floor(log_2(p-2^q)),

F(x, r) = floor(x/2^q)*2^q + Sum_{k = 0...q - 1} (floor(x/2^(r-k)) mod 2)*2^k;

If F(p, m) <= n then a(n) = F(p, m), otherwise a(n) = F(p-2^q, s).

By definition: F(p, m) = floor(p/2^q)*2^q + A030101(p) mod 2^q; also: F(p-2^q, s) = floor((p-2^q)/2^q)*2^q + A030101(p-2^q) mod 2^q; [Edited and corrected by Hieronymus Fischer, Sep 08 2018]

a(n) = A006995(A206915(n));

a(n) = A006995(A206915(A206914(n+1))-1);

a(n) = A006995(A206916(A206914(n+1))-1).

EXAMPLE

a(0) = 0 since 0 is the greatest binary palindrome <= 0;

a(1) = 1 since 1 is the greatest binary palindrome <= 1;

a(2) = 1 since 1 is the greatest binary palindrome <= 2;

a(3) = 3 since 3 is the greatest binary palindrome <= 3.

PROG

(Haskell)

a206913 n = last $ takeWhile (<= n) a006995_list

-- Reinhard Zumkeller, Feb 27 2012

CROSSREFS

Cf. A006995, A206915, A206920, A030301.

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Sequence in context: A129337 A133909 A177691 * A117767 A340836 A293701

Adjacent sequences: A206910 A206911 A206912 * A206914 A206915 A206916

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Feb 13 2012

STATUS

approved