cp's OEIS Frontend

A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

%I A063686 #17 Sep 04 2025 08:34:16
%S A063686 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,4,4,2,1,1,1,1,4,6,4,2,1,
%T A063686 1,1,1,7,11,8,4,2,1,1,1,1,9,19,14,8,4,2,1,1,1,1,14,33,27,16,8,4,2,1,1,
%U A063686 1,1,18,56,50,30,16,8,4,2,1,1,1,1,30,101,96,59,32,16,8,4,2,1,1,1
%N A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.
%C A063686 Column k=1 appears to be A032190(n), n=2,3,...
%H A063686 Andrew Howroyd, <a href="/A063686/b063686.txt">Table of n, a(n) for n = 0..1274</a> (Rows n=0..49)
%e A063686 Triangle begins:
%e A063686   1;
%e A063686   1, 1;
%e A063686   1, 1, 1;
%e A063686   1, 1, 1, 1;
%e A063686   1, 2, 1, 1, 1;
%e A063686   1, 2, 2, 1, 1, 1;
%e A063686   1, 4, 4, 2, 1, 1, 1;
%e A063686   1, 4, 6, 4, 2, 1, 1, 1;
%e A063686   1, 7, 11, 8, 4, 2, 1, 1, 1;
%e A063686   1, 9, 19, 14, 8, 4, 2, 1, 1, 1;
%e A063686   1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1;
%e A063686   ...
%o A063686 (PARI) \\ here R(n) is A048887 transposed
%o A063686 R(n)={Mat(vector(n, k, Col((1-x)/(1-2*x+x^(k+1)) - 1 + O(x*x^n))))}
%o A063686 S(M)={matrix(#M-1, #M-1, n, k, if(k<n-1, (k+1)*M[n-k-1,k+1] + sum(j=1, k, j*(M[n-j,k+1]-M[n-j,k])), if(k<n, n)))}
%o A063686 T(n)={my(M=S(R(n+1))); matid(n) + matrix(n, n, n, k, sumdiv(n, d, if(k<d, eulerphi(n/d)*M[d, k]))/n) }
%o A063686 my(M=T(10)); for(n=0, #M, for(k=0, n, print1(if(k==0,1,M[n,k]), ", ")); print) \\ _Andrew Howroyd_, Oct 15 2017
%Y A063686 Cf. A032190, A048004, A048887.
%Y A063686 Cf. A000358, A093305, A280218 (necklaces avoiding 00, 000, 0000).
%K A063686 nonn,tabl,changed
%O A063686 0,12
%A A063686 Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 22 2001
%E A063686 T(0,0)=1 from _Andrew Howroyd_, Oct 15 2017