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%I #29 Nov 15 2015 13:31:37
%S 1,2,4,6,7,9,10,12,13,15,16,18,19,21,23,24,26,28,29,31,33,34,36,38,39,
%T 41,43,44,46,48
%N Maximal number of palindromes in a circular binary word of length n.
%C The word is to be imagined as written around a circle. We only count palindromes of length m with 1 <= m <= n.
%H David Consiglio, Jr., <a href="/A247000/a247000.py.txt">Alternate Python Program</a>
%H Jamie Simpson, <a href="http://dx.doi.org/10.1016/j.tcs.2014.07.012">Palindromes in circular words</a>, Theoretical Computer Science, Volume 550, 18 September 2014, Pages 66-78; DOI: 10.1016/j.tcs.2014.07.012.
%F Conjectures from _Colin Barker_, Nov 14 2015: (Start)
%F a(n) = a(n-1)+a(n-3)-a(n-4) for n>4.
%F G.f.: x*(x^13-x^12+x^11-x^10+x^9-x^8+x^7-x^6+x^3+2*x^2+x+1) / ((x-1)^2*(x^2+x+1)).
%F (End)
%e n a(n) Example of a word with a(n) palindromes
%e 1 1 (a)
%e 2 2 (aa)
%e 3 4 (aab)
%e 4 6 (aabb) Palindromes are a,b,aa,bb,abba,baab
%e 5 7 (aaaab)
%e 6 9 (aaaabb)
%e 7 10 (aaaaaab)
%e 8 12 (aaaaaabb)
%e 9 13 (aaaaaaaab)
%e 10 15 (aaaaaaaabb)
%e 11 16 (aaaaaaaaaab)
%e 12 18 (aaaaaaaaaabb)
%e 13 19 (aaaaaaaaaaaab)
%e 14 21 (aaaaaaaaaaaabb)
%e 15 23 (aaaaabaaaabaaab)
%e 16 24 (aaaaaaaaaaaaaabb)
%e 17 26 (aaaaaabaaaaabaaab)
%e 18 28 (aaaaaabaaaaabaaaab)
%e 19 29 (aaaaaaabaaaaaabaaab)
%e 20 31 (aaaaaaabaaaaaabaaaab)
%e 21 33 (aaaaaaabaaaaaabaaaaab)
%e 22 34 (aaaaaaaabaaaaaaabaaaab)
%e 23 36 (aaaaaaaabaaaaaaabaaaaab)
%e 24 38 (aaaaaaaabaaaaaaabaaaaaab)
%e 25 39 (aaaaaaaaabaaaaaaaabaaaaab)
%e 26 41 (aaaaaaaaabaaaaaaaabaaaaaab)
%e 27 43 (aaaaaaaaabaaaaaaaabaaaaaaab)
%e 28 44 (aaaaaaaaaabaaaaaaaaabaaaaaab)
%e 29 46 (aaaaaaaaaabaaaaaaaaabaaaaaaab)
%e 30 48 (aaaaaaaaaabaaaaaaaaabaaaaaaaab)
%o (Python)
%o def A247000(n):
%o ....maxcount = 0
%o ....for i in range(2**(n-1),2**n):
%o ........s = format(i,'0'+str(n)+'b')
%o ........s, plist = s+s[:-1], []
%o ........for j in range(n):
%o ............for k in range(n):
%o ................t = s[j:j+k+1]
%o ................if t == t[::-1] and not t in plist:
%o ....................plist.append(t)
%o ........if len(plist) > maxcount:
%o ............maxcount = len(plist)
%o ....return maxcount # _Chai Wah Wu_, Sep 16 2014
%K nonn,more
%O 1,2
%A _N. J. A. Sloane_, Sep 15 2014
%E More terms from _David Consiglio, Jr._, Sep 16 2014