A206927 Minimal numbers of binary length n+1 such that the number of contiguous palindromic bit patterns in the binary representation is minimal.

2, 4, 9, 18, 37, 75, 150, 300, 601, 1202, 2405, 4811, 9622, 19244, 38489, 76978, 153957, 307915, 615830, 1231660, 2463321, 4926642, 9853285, 19706571, 39413142, 78826284, 157652569, 315305138, 630610277, 1261220555

Offset

1,1

Comments

Subsequence of A206926.

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.

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

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,1,-2).

Formula

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

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

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

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

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

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

Example

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.
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.

Crossrefs

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

Sequence in context: A182028 A081253 A118255 * A019299 A052932 A018097

Adjacent sequences: A206924 A206925 A206926 * A206928 A206929 A206930

Keyword

nonn,base

Author

Hieronymus Fischer, Mar 24 2012

Extensions

Further formulas added by Hieronymus Fischer, Jan 13 2013

Status

approved