login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Number of binary strings of length n having a cyclic shift that is a palindrome.
1

%I #29 Mar 11 2017 14:46:50

%S 1,2,2,8,6,32,20,100,54,260,152,684,348,1640,884,3728,1974,8672,4556,

%T 19420,10056,42736,22508,94164,48636,204632,106472,441764,228444,

%U 950216,491120,2031556,1046454,4323888,2228192,9174400,4713252,19398584

%N Number of binary strings of length n having a cyclic shift that is a palindrome.

%C Alternatively (Luke Schaeffer), those binary strings of length n that can be written as the concatenation of two palindromes, one of which has even length. - _Jeffrey Shallit_, Jan 28 2016

%C Apparently A045655 is a bisection. - _R. J. Mathar_, Mar 11 2017

%H Lars Blomberg, <a href="/A245582/a245582.txt">C# program for generating the sequence.</a>

%e For n = 4 the function counts the strings 0000, 0011, 0110 and their bitwise complements.

%t palQ[lst_] := lst==Reverse[lst]; a[n_] := (For[k=0; cnt=0, k<2^n, k++, bb = PadLeft[IntegerDigits[k, 2], n]; If[MatchQ[bb, {a___ /; palQ[{a}], b___ /; palQ[{b}]} /; EvenQ[Length[{a}]] || EvenQ[Length[{b}]]], cnt++]]; cnt); Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* _Jean-François Alcover_, Jan 28 2016 *)

%K nonn

%O 0,2

%A _Jeffrey Shallit_, Jul 26 2014

%E a(25)-a(37) from _Lars Blomberg_, Oct 20 2014