A039791 Sequence arising in search for Legendre sequences.

1, 1, 2, 4, 6, 14, 66, 95, 280, 1464, 2694, 10452, 41410, 95640, 323396, 1770963, 5405026, 13269146, 73663402, 164107650, 582538732, 3811895344, 7457847082, 30712068524, 151938788640, 353218528324, 1738341231644, 7326366290632, 17280039555348, 63583110959728

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

Table of n, a(n) for n=1..30.

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

Coincides with A002995 offset by -1 at the A005097-th terms.

Cf. A000108, A007123, A001700.

Sequence in context: A080198 A077637 A077639 * A275664 A349979 A192536

Adjacent sequences: A039788 A039789 A039790 * A039792 A039793 A039794

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Travis Scott, Nov 24 2022

Status

approved