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%I #42 Jan 31 2023 11:53:25
%S 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,
%T 13,11,11,11,10,10,11,11,10,12,11,10,11,11,13,13,11,10,11,10,11,11,12,
%U 12,11,11,12,13,13,16,21,22,18,15,15,14,13,13,14,14
%N Number of contiguous palindromic bit patterns in the binary representation of n.
%C 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).
%C 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.
%H Reinhard Zumkeller, <a href="/A206925/b206925.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>
%F a(n) <= (m+1)*(m+2)/2, where m = floor(log_2(n)); equality holds if n + 1 is a power of 2.
%F a(n) >= 2*floor(log_2(n)).
%F This estimation cannot be improved in general, since equality holds for A206926(n): a(A206926(n)) = 2*floor(log_2(A206926(n))).
%F Asymptotic behavior:
%F a(n) = O(log(n)^2).
%F lim sup a(n)/log_2(n)^2 = 1/2, for n --> infinity.
%F lim inf a(n)/log_2(n) = 2, for n --> infinity.
%e a(1) = 1, since 1 = 1_2 is the only palindromic bit pattern;
%e a(4) = 4, since 4 = 100_2 and there are the following palindromic bit patterns: 1, 0, 0, 00;
%e a(5) = 4, since 5 = 101_2 and there are the following palindromic bit patterns: 1, 0, 1, 101;
%e a(8) = 7, since 8 = 1000_2 and there are the following palindromic bit patterns: 1, 0, 0, 0, 00, 00, 000.
%o (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
%o (Haskell)
%o import Data.Map (fromList, (!), insert)
%o import Data.List (inits, tails)
%o a206925 n = a206925_list !! (n-1)
%o a206925_list = 1 : f [0, 1] (fromList [(Bin [0], 1), (Bin [1], 1)]) where
%o f bs'@(b:bs) m = y : f (succ bs') (insert (Bin bs') y m) where
%o y = m ! (Bin bs) +
%o length (filter (\ds -> ds == reverse ds) $ tail $ inits bs')
%o succ [] = [1]; succ (0:ds) = 1 : ds; succ (1:ds) = 0 : succ ds
%o -- _Reinhard Zumkeller_, Dec 17 2012
%o (Smalltalk)
%o A206925
%o "Answers the number of symmetric bit patterns of n as a binary."
%o | m p q n numSym |
%o n := self.
%o n < 2 ifTrue: [^1].
%o m := n integerFloorLog: 2.
%o p := n printStringRadix: 2.
%o numSym := 0.
%o 1 to: m + 1
%o do:
%o [:k |
%o 1 to: k
%o do:
%o [:j |
%o q := p copyFrom: j to: k.
%o q = q reverse ifTrue: [numSym := numSym + 1]]].
%o ^numSym // _Hieronymus Fischer_, Feb 16 2013
%o (Python)
%o def A206925(n):
%o s = bin(n)[2:]
%o k = len(s)
%o 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
%Y Cf. A006995, A206923, A206924, A206925, A206926, A070939.
%Y Cf. A215244, A030308.
%K nonn,base
%O 1,2
%A _Hieronymus Fischer_, Mar 12 2012
%E Comments and formulas added by _Hieronymus Fischer_, Jan 23 2013
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