A206925 Number of contiguous palindromic bit patterns in the binary representation of n.

1, 2, 3, 4, 4, 4, 6, 7, 6, 6, 6, 6, 6, 7, 10, 11, 9, 8, 8, 8, 9, 8, 9, 9, 8, 8, 9, 9, 9, 11, 15, 16, 13, 11, 11, 11, 10, 10, 11, 11, 10, 12, 11, 10, 11, 11, 13, 13, 11, 10, 11, 10, 11, 11, 12, 12, 11, 11, 12, 13, 13, 16, 21, 22, 18, 15, 15, 14, 13, 13, 14, 14

Offset: 1

Author

Hieronymus Fischer, Mar 12 2012

Keywords

• nonn
• base

Comments

The number of contiguous palindromic bit patterns in the binary representation of n is a measure for the grade of symmetry in an abstract arrangement of two kinds of elements (where the number of elements is the number of binary digits, of course).

The minimum value for a(n) is 2*floor(log_2(n)) and will be taken infinitely often (see A206926 and A206927). This means: For a given number of places m there are at least 2*(m-1) palindromic substrings in the binary representation. This lower bound indicates to a certain extent the minimal possible symmetry.

Examples

			a(1) = 1, since 1 = 1_2 is the only palindromic bit pattern;
a(4) = 4, since 4 = 100_2 and there are the following palindromic bit patterns: 1, 0, 0, 00;
a(5) = 4, since 5 = 101_2 and there are the following palindromic bit patterns: 1, 0, 1, 101;
a(8) = 7, since 8 = 1000_2 and there are the following palindromic bit patterns: 1, 0, 0, 0, 00, 00, 000.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to palindromes

Crossrefs

Cf. A006995, A206923, A206924, A206925, A206926, A070939.

Cf. A215244, A030308.

Programs

Haskell

import Data.Map (fromList, (!), insert)
import Data.List (inits, tails)
a206925 n = a206925_list !! (n-1)
a206925_list = 1 : f [0, 1] (fromList [(Bin [0], 1), (Bin [1], 1)]) where
   f bs'@(b:bs) m = y : f (succ bs') (insert (Bin bs') y m) where
     y = m ! (Bin bs) +
         length (filter (\ds -> ds == reverse ds) $ tail $ inits bs')
     succ [] = [1]; succ (0:ds) = 1 : ds; succ (1:ds) = 0 : succ ds
-- Reinhard Zumkeller, Dec 17 2012

PARI

a(n)=n=binary(n);sum(k=0,#n-1,sum(i=1,#n-k,prod(j=0, k\2,n[i+j]==n[i+k-j]))) \\ Charles R Greathouse IV, Mar 21 2012

Python

def A206925(n):
    s = bin(n)[2:]
    k = len(s)
    return sum(1 for i in range(k) for j in range(i+1,k+1) if s[i:j] == s[j-1:i-1-k:-1]) # Chai Wah Wu, Jan 31 2023

Smalltalk

A206925
"Answers the number of symmetric bit patterns of n as a binary."
| m p q n numSym |
n := self.
n < 2 ifTrue: [^1].
m := n integerFloorLog: 2.
p := n printStringRadix: 2.
numSym := 0.
1 to: m + 1
  do:
   [:k |
   1 to: k
    do:
     [:j |
     q := p copyFrom: j to: k.
     q = q reverse ifTrue: [numSym := numSym + 1]]].
^numSym // Hieronymus Fischer, Feb 16 2013

Formula

a(n) <= (m+1)*(m+2)/2, where m = floor(log_2(n)); equality holds if n + 1 is a power of 2.

a(n) >= 2*floor(log_2(n)).

This estimation cannot be improved in general, since equality holds for A206926(n): a(A206926(n)) = 2*floor(log_2(A206926(n))).

Asymptotic behavior:

a(n) = O(log(n)^2).

lim sup a(n)/log_2(n)^2 = 1/2, for n --> infinity.

lim inf a(n)/log_2(n) = 2, for n --> infinity.

Extensions

Comments and formulas added by Hieronymus Fischer, Jan 23 2013