A371570 -id:A371570 - cp's OEIS frontend

A371668 Number of binary necklaces of length n which have more 00 than 01 substrings.

0, 1, 1, 1, 5, 11, 19, 43, 93, 181, 371, 771, 1547, 3121, 6357, 12821, 25805, 52123, 105031, 211243, 425215, 855457, 1719257, 3455153, 6942387, 13942111, 27993317, 56197117, 112785797, 226311535, 454043339, 910778203, 1826666093, 3663122277, 7344953123

Offset: 0

Robert P. P. McKone, Apr 02 2024

nonn

			a(3) = 1: 000.
a(4) = 5: 0000, 0001, 0010, 0100, 1000.
a(5) = 11: 00000, 00001, 00010, 00011, 00100, 00110, 01000, 01100, 10000, 10001, 11000.

Cf. A217464 (necklaces with equal 00 and 01), A371570 (necklaces with more 01 than 00).
Cf. A126869 (necklaces with equal 00 and 11, for n>=1), A058622 (necklaces with more 00 than 11).
Cf. A163493 (strings with equal 00 and 01), A371358 (strings with more 00 than 01), A371564 (strings with more 01 than 00).

tup[n_] := Tuples[{0, 1}, n];
tupToNec[n_] := Map[Append[#, #[[1]]] &, tup[n]];
cou[lst_List] := Count[lst, {0, 0}] > Count[lst, {0, 1}];
par[lst_List] := Partition[lst, 2, 1];
a[0] = 0;
a[n_] := a[n] = Map[cou, Map[par, tupToNec[n]]] // Boole // Total;
Monitor[Table[a[n], {n, 0, 18}], {n, Table[a[m], {m, 0, n - 1}]}]

a(n) = 2^n - A217464(n) - A371570(n).

a(n) = (8*(n-7)*a(n-7) + 4*(29-5*n)*a(n-6) + (26*n-110)*a(n-5) + (77-23*n)*a(n-4) + (15*n-46)*a(n-3) + (22-10*n)*a(n-2) + 5*(n-1)*a(n-1))/n for n>=7.

cp's OEIS Frontend

A371668 Number of binary necklaces of length n which have more 00 than 01 substrings.

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