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 A332441 #46 Feb 11 2021 09:23:54
%S A332441 6,10,42,54,110,78,60,68,342,42,506,250,486,406,310,330,420,666,156,
%T A332441 410,602,540,2162,2058,408,1378,220,342,3422,1830,378,390,4422,1518,
%U A332441 4970,1314,1500,2310,6162,4374,6806,680,2436,1958,1092,930,3420,2328,2970,5050,10506
%N A332441 The lengths of the primitive periods of the partial sums of the periodic unsigned Schick sequences with initial value 1, for N = 2*n + 1, for n >= 1, taken modulo 2*N.
%C A332441 For the signed Schick sequences see the Schick reference, where the odd N is named p. The unsigned Schick sequences are used in the Brändli and Beyne paper.
%C A332441 See also a comment in A332439 where the periodic unsigned Schick sequences are named SBBseq(N, q0), with B(N) = A135303((N-1)/2) different odd initial values q0 satisfying gcd(q0, N) = 1. The complete set of the primitive periods SBB(N, q0) of these sequences is named SBB(N).
%C A332441 The length of the primitive periods SBB(N, q0) is identical for each of the B(N) different q0 values, and named pes(N) by Schick.
%C A332441 Here only the lengths of the primitive periods of the partial sums of SBBseq(N, q0 = 1) (mod 2*N) is given, namely a(n) = L(2*n+1, 1).
%C A332441 Note that this length depends in general on the initial value q0: L(2*n+1, q0). For example, the B(65) = 4 initial values q0 = 1, 3, 7, and 11 for n = 32, N = 65, have lengths a(32) = 390, 390, 78 = 390/5, and 390, respectively.
%C A332441 The general length formula is L(N, q0) = 2*N*pes(N)/gcd(SUM(SBB(N, q0)), 2*N), with pes(N) = A003558((N-1)/2), and the gcd values are shown for the N values with B(N) = 1 (q0 = 1) in A333849, and for more than one initial value (B(N) >= 2) in A333851.
%C A332441 a(n) gives also the length of the corresponding Euler tour ET(2*n+1, q0 = 1), which may not involve all vertices of a regular (2*(2*n+1))-gon. Also the digraphs underlying these Euler tours are not always regular. See some examples below.
%D A332441 Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.
%H A332441 Gerold Brändli and Tim Beyne, <a href="https://arxiv.org/abs/1504.02757">Modified Congruence Modulo n with Half the Amount of Residues</a>, arXiv:1504.02757 [math.NT], 2015-2016.
%F A332441 The length a(n) = L(2*n+1 = N) = Sum_{j=1..Neff(N)} degree(Veff^{(2*N)}(j))/2, where Neff(N) is the number of vertices Veff^{(2*N)}, which are visited by the Euler tour. See the example N = 21 with Neff = 21 (not 2*N = 42) below.
%F A332441 a(n) = L(2*n+1 = N) = 2*N*A003558((N-1)/2)/A333849((N-1)/2), except for those N values from A333855 with the denominator replaced by the first gcd value given in the rows of array A333851. See a comment above for the general L(N, q0) formula.
%e A332441 n = 8 (N = 17): B(17) = 2, pes(17) = 4. SBBseq(17, 1) = repeat(1, 15, 13, 9, ), SBBseq(17, 3) = repeat(3, 11, 5, 7, ). Euler tour ET(N, 1) = [0, 1, 16, 29, 4, 5, 20, 33, 8, 9, 24, 3, 12, 13, 28, 7, 16, 17, 32, 11, 20, 21, 2, 15, 24, 25, 6, 19, 28, 29, 10, 23, 32, 33, 14, 27, 2, 3, 18, 31, 6, 7, 22, 1, 10, 11, 26, 5, 14, 15, 30, 9, 18, 19, 0, 13, 22, 23, 4, 17, 26, 27, 8, 21, 30, 31, 12, 25, 0]. This corresponds to a regular digraph of degree 4. Neff(17) = 2*17 = 34, L(17) = 34*4/2 = 68 = a(8). Note that for N = 17 the denominator is A333851(1, 1) = 2. There is another Euler tour ET(N, 2) of the same length.
%e A332441 n = 10 (N = 21): B(21) = 1, pes(21) = 6. SBBseq(21, 1) = repeat(1, 19, 17, 13, 5, 11, ). The Euler tour ET(N, 1) = [0, 1, 20, 37, 8, 13, 24, 25, 2, 19, 32, 37, 6, 7, 26, 1, 14, 19, 30, 31, 8, 25, 38, 1, 12, 13, 32, 7, 20, 25, 36, 37, 14, 31, 2, 7, 18, 19, 38, 13, 26, 31, 0]. The Neff(21) = 21 vertex labels for the 42-gon are {6*k, 6*k+1, 6*k+2}, for k = 0..6. The digraph is not regular, the vertices with labels 6*k have degree 2 (visited once), for labels 6*k+1 the degree is 6, and for labels 6*k+2 the degree is 4. All other 21 vertices of the 42-gon are not involved (or have degree 0, and the connectivity number of the unconnected digraph is 22). L(21) = 7*(2/2 + 6/2 + 4/2) = 7*6 = 42 = a(10) = 2*21*6/6, because A333849(10) = 6.
%o A332441 (PARI) A333848(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m])));
%o A332441 A333849(n) = gcd(A333848(n), 2*(2*n+1));
%o A332441 isok8(m, n) = my(md = Mod(2, 2*n+1)^m); (md==1) || (md==-1);
%o A332441 A003558(n) = my(m=1); while(!isok8(m, n) , m++); m;
%o A332441 B(n) = eulerphi(n)/(2*A003558((n-1)/2));
%o A332441 a(n) = {my(m = 2*n+1, period = A003558(n)); if (B(m) == 1, return(2*m*period/A333849(n))); my(q=1, qs = List([q])); for (i=1, period-1, q = abs(m-2*q); listput(qs, q);); 2*m*period/gcd(vecsum(Vec(qs)), 2*m);} \\ _Michel Marcus_, Jun 14 2020
%Y A332441 Cf. A003558, A135303, A332439, A333849, A333850, A333851.
%K A332441 nonn
%O A332441 1,1
%A A332441 _Wolfdieter Lang_, Apr 04 2020
%E A332441 More terms from _Michel Marcus_, Jun 14 2020