This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A039791 #37 Dec 02 2022 13:14:58
%S A039791 1,1,2,4,6,14,66,95,280,1464,2694,10452,41410,95640,323396,1770963,
%T A039791 5405026,13269146,73663402,164107650,582538732,3811895344,7457847082,
%U A039791 30712068524,151938788640,353218528324,1738341231644,7326366290632,17280039555348,63583110959728
%N A039791 Sequence arising in search for Legendre sequences.
%C A039791 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
%H A039791 Roderick J. Fletcher, Marc Gysin, and Jennifer Seberry, <a href="http://ajc.maths.uq.edu.au/pdf/23/ajc-v23-p75.pdf">Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices</a>, Australasian J. Combin. 23 (2001), 75-86.
%F A039791 a(n) ~ C(2n+1, n)/(2n+1)/phi(2n+1)
%F A039791 Empirical: a(n) == 1 (mod 2) for 2n+1 of the form 2^k+1 but not of the form p^2, else == 0.
%e A039791 From _Travis Scott_, Nov 24 2022: (Start)
%e A039791 If we decompose by weight the classes of period 2n+1 counted at A002729, a(n) appears as the twin towers of that triangle.
%e A039791 a(n)
%e A039791 | |
%e A039791 (1) (1)
%e A039791 1 1 1 1
%e A039791 1 1 1 1 1 1
%e A039791 1 1 1 2 2 1 1 1
%e A039791 1 1 2 3 4 4 3 2 1 1
%e A039791 1 1 1 2 4 6 6 4 2 1 1 1
%e A039791 1 1 1 3 7 10 14 14 10 7 3 1 1 1
%e A039791 1 1 3 7 18 34 54 66 66 54 34 18 7 3 1 1
%e A039791 1 1 1 3 11 25 49 75 95 95 75 49 25 11 3 1 1 1. (End)
%t A039791 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 *)
%Y A039791 Coincides with A002995 offset by -1 at the A005097-th terms.
%Y A039791 Cf. A000108, A007123, A001700.
%K A039791 nonn
%O A039791 1,3
%A A039791 _N. J. A. Sloane_
%E A039791 More terms from _Travis Scott_, Nov 24 2022