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A371570 Number of binary necklaces of length n which have more 01 than 00 substrings.

%I A371570 #20 Apr 02 2024 11:44:03
%S A371570 0,0,2,3,6,15,29,56,118,237,467,946,1905,3796,7618,15303,30614,61319,
%T A371570 122951,246202,492971,987542,1977560,3959289,7927969,15873190,
%U A371570 31776708,63614397,127346134,254908115,510233309,1021273672,2044071894,4091064805,8187770675
%N A371570 Number of binary necklaces of length n which have more 01 than 00 substrings.
%C A371570 A necklace may also be referred to as circular or cyclic strings.
%F A371570 a(n) = 2^n - A217464(n) - A371668(n).
%F A371570 a(n) = -(((n-3)*(n-2) - 8*(n-5)^2*(n-2)*a(n-5) + 4*(n*((3n-34)*n+117)-114)*a(n-4) + 2*(((32-3n)*n-95)*n+62)*a(n-3) + (((5n-52)*n+157)*n-114)*a(n-2) + (((39-4n)*n-103)*n+58)*a(n-1))/((n-6)*(n-3)*n)) for n>=7.
%e A371570 a(3) = 3: 011, 101, 110.
%e A371570 a(4) = 6: 0101, 0111, 1010, 1011, 1101, 1110.
%e A371570 a(5) = 15: 00101, 01001, 01010, 01011, 01101, 01111, 10010, 10100, 10101, 10110, 10111, 11010, 11011, 11101, 11110.
%t A371570 tup[n_] := Tuples[{0, 1}, n];
%t A371570 tupToNec[n_] := Map[Append[#, #[[1]]] &, tup[n]];
%t A371570 cou[lst_List] := Count[lst, {0, 1}] > Count[lst, {0, 0}];
%t A371570 par[lst_List] := Partition[lst, 2, 1];
%t A371570 a[0] = 0;
%t A371570 a[n_] := Map[cou, Map[par, tupToNec[n]]] // Boole // Total;
%t A371570 Monitor[Table[a[n], {n, 0, 18}], {n, Table[a[m], {m, 0, n - 1}]}]
%Y A371570 Cf. A217464 (necklaces with equal 00 and 01), A371668 (necklaces with more 00 than 01).
%Y A371570 Cf. A126869 (necklaces with equal 00 and 11, for n>=1), A058622 (necklaces with more 00 than 11).
%Y A371570 Cf. A163493 (strings with equal 00 and 01), A371358 (strings with more 00 than 01), A371564 (strings with more 01 than 00).
%K A371570 nonn
%O A371570 0,3
%A A371570 _Robert P. P. McKone_, Mar 28 2024