A206927 - OEIS

%I #27 Mar 19 2024 08:31:23

%S 2,4,9,18,37,75,150,300,601,1202,2405,4811,9622,19244,38489,76978,

%T 153957,307915,615830,1231660,2463321,4926642,9853285,19706571,

%U 39413142,78826284,157652569,315305138,630610277,1261220555

%N Minimal numbers of binary length n+1 such that the number of contiguous palindromic bit patterns in the binary representation is minimal.

%C Subsequence of A206926.

%C From left to right, the binary representation of a(n) consists of a concatenation of the bit pattern 100101 (=37). If the number of places is not a multiple of 6, the least significant places are truncated. This leads to a simple linear recurrence.

%C Example: a(19)=615830=10010110010110_2=concatenate('100101','100101','10')

%H Hieronymus Fischer, <a href="/A206927/b206927.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,0,1,-2).

%F a(n) = 37*2^(1+n mod 6)*(2^(6*floor(n/6))-1)/63 + floor(37*2^(n mod 6)/2^5).

%F a(n) = floor((37*2^(n+1)/63)) mod 2^(n+1).

%F A206925(a(n)) = 2*floor(log_2(a(n))).

%F a(n+1) = 2a(n) + floor(37*2^(n+2)/63) mod 2.

%F G.f.: x*( 2+x^2+x^4+x^5-2*x^6 ) / ( (x-1)*(2*x-1)*(1+x)*(x^2-x+1)*(1+x+x^2) ). - R. J. Mathar, Apr 02 2012

%F Also, g.f. x*(2+x^2+x^4+x^5-2*x^6)/((1-2*x)*(1-x^6)).

%e a(3)=9=1001_2 has 6 [=A206925(9)] contiguous palindromic bit patterns. This is the minimum value for binary numbers with 4 places and 9 is the least number with this property.

%e a(9)=601=1001011001_2 has 18 [=A206925(601)] contiguous palindromic bit patterns. This is the minimum value for binary numbers with 10 places and 601 is the least number with this property.

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

%K nonn,base

%O 1,1

%A _Hieronymus Fischer_, Mar 24 2012

%E Further formulas added by _Hieronymus Fischer_, Jan 13 2013