OFFSET
1,3
COMMENTS
Number of bit strings of length L = 2n+1 and Hamming weight n (or n+1, as generated by Fletcher et al.) up to chord equivalence (i.e., up to color and general linear permutation x -> Ax+b mod L for A on Z/LZ* and b on Z/LZ--essentially a multiplicative necklace of phi(L) additive necklaces of L black and white beads where L is odd and the colors are as balanced as possible). The same strings are counted up to bracelet equivalence (x -> +-x+b mod L) at A007123, up to necklace equivalence (x -> x+b mod L) at A000108, and in full (x -> x) at A001700. - Travis Scott, Nov 24 2022
LINKS
Roderick J. Fletcher, Marc Gysin, and Jennifer Seberry, Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices, Australasian J. Combin. 23 (2001), 75-86.
FORMULA
a(n) ~ C(2n+1, n)/(2n+1)/phi(2n+1)
Empirical: a(n) == 1 (mod 2) for 2n+1 of the form 2^k+1 but not of the form p^2, else == 0.
EXAMPLE
From Travis Scott, Nov 24 2022: (Start)
If we decompose by weight the classes of period 2n+1 counted at A002729, a(n) appears as the twin towers of that triangle.
a(n)
| |
(1) (1)
1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 1 1 1
1 1 2 3 4 4 3 2 1 1
1 1 1 2 4 6 6 4 2 1 1 1
1 1 1 3 7 10 14 14 10 7 3 1 1 1
1 1 3 7 18 34 54 66 66 54 34 18 7 3 1 1
1 1 1 3 11 25 49 75 95 95 75 49 25 11 3 1 1 1. (End)
MATHEMATICA
Module[{a, b, g, L, m, x, z, Z}, Table[L=2n+1; Z=Sum[Sum[Product[g=L/GCD[L, (k-1)i+j]; Subscript[x, #]^(1/#)&@If[k==1, g, m=MultiplicativeOrder[k, g]; g/GCD[g, (k^m-1)/(k-1)]m], {i, L}]L/GCD[L, k-1], {j, GCD[L, k-1]}], {k, Select[Range@L, CoprimeQ[#, L]&]}]/L/EulerPhi@L/.Subscript[x, z_]->a^z+b^z; CoefficientList[Z, {a, b}][[n+1, n+2]], {n, 30}]] (* Travis Scott, Nov 24 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Travis Scott, Nov 24 2022
STATUS
approved