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A206927 Minimal numbers of binary length n+1 such that the number of contiguous palindromic bit patterns in the binary representation is minimal. 4
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 (list; graph; refs; listen; history; text; internal format)
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 - 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

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Last modified March 26 00:27 EDT 2019. Contains 321479 sequences. (Running on oeis4.)