A056366
Number of primitive (period n) bracelet structures using exactly two different colored beads.
Original entry on oeis.org
0, 1, 1, 2, 3, 5, 8, 14, 21, 39, 62, 112, 189, 352, 607, 1144, 2055, 3885, 7154, 13602, 25472, 48670, 92204, 176770, 337590, 649341, 1246840, 2404872, 4636389, 8964143, 17334800
Offset: 1
- M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
A364468
Number of primitive n-bead necklaces (turning over is allowed) comprising elements of two flavors where complements and scalings are equivalent.
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 3, 8, 12, 20, 35, 62, 106, 189, 343, 603, 1130, 2055, 3860, 7154, 13562, 25463, 48607, 92204, 176646, 337587, 649151, 1246819, 2404519, 4636389, 8963497, 17334800, 33585928, 65107935, 126385919, 245492221, 477345359, 928772649, 1808662015, 3524337599, 6872457828, 13409202675, 26179870365
Offset: 0
For a(4) = 1, there is one solution: "1110". The other primitive sequence "1100" can be reduced to "10", which no longer uses 4 elements.
For a(6) = 3, there are three solutions: "111110", "111010", and "110010". The other primitive sequences "111100" and "111000" can be reduced to "110" and "10", respectively, which no longer use 6 elements.
-
a11(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n));
a46(n) = {
my(s=0);
fordiv (n, d,
s+=moebius(d)*a11(n/d));
s};
a364468(n) = {
my(s=a46(n));
fordiv (n, k,
s-=if(k!=1&&k!=n, a364468(k), 0));
s};
for (k=1,42, print1(a364468(k),", ")) \\ Hugo Pfoertner, Jul 26 2023
Comments