cp's OEIS Frontend

Counts contiguously substituted cycloalkane polyols (CSCPs).

From Ed Wynn and Andrew Howroyd, May 22 2021: (Start)

Consider a bracelet of n beads, each colored blue on one side and red on the other. Turning the bracelet over has the effect of simultaneously swapping the colors and reversing the order of the beads. For example, rrbbrb when turned over becomes rbrrbb. The total number of such bracelets is counted by A053656(n).

Swapping the colors is equivalent to reversing the order of the beads. For example, rrbbrb becomes bbrrbr which when turned over is brbbrr. A bracelet may or may not be the same as its reversal (or complement). The case of equality is counted by A256217(n) and the remainder can be divided into "chiral" pairs which are the reverse of each other and counted by this sequence. a(n) is then the number of pairs of two-colored n-bead bracelets that are equal under reversal but unequal up to rotation and turning over.

In chemical terms, these pairs are called "enantiomeric pairs". The example of rrbbrb corresponds to a pair of "chiral" chemical molecules: L-chiro-inositol and R-chiro-inositol.

a(n) is also half the number of nonisomorphic orientations of the n-cycle graph which are not self-converse. Again the self-converse orientations are counted by A256217(n) and the total by A053656(n).

(End)

Counts contiguously substituted cycloalkane polyols (CSCPs).

Consider a bracelet of n beads, each colored blue on one side and red on the other -- each bead changes color when the bracelet is turned over. Reversal is then the same as swapping the colors. a(n) is the number of colorings that are invariant under reversal, up to rotation and turning over. - Ed Wynn, May 22 2021

Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.

This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit and Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix, Nov 17 2005

Except for a(0) = 1, Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 7th (rightmost) column. Other columns of the table include (but are not identified as) A006206-A006208. - Jonathan Vos Post, Jun 18 2007

"Number of binary Lyndon words" means: number of binary strings inequivalent modulo rotation (cyclic permutation) of the digits and not having a period smaller than n. This provides a link to A103314, since these strings correspond to the inequivalent zero-sum subsets of U_m (m-th roots of unity) obtained by taking the union of U_n (n|m) with 0 or more U_d (n | d, d | m) multiplied by some power of exp(i 2Pi/n) to make them mutually disjoint. (But not all zero-sum subsets of U_m are of that form.) - M. F. Hasler, Jan 14 2007

Also the number of dynamical cycles of period n of a threshold Boolean automata network which is a quasi-minimal positive circuit of size a multiple of n and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Feb 25 2009

Also, the number of periodic points with (minimal) period n in the iteration of the tent map f(x):=2min{x,1-x} on the unit interval. - Pietro Majer, Sep 22 2009

Number of distinct cycles of minimal period n in a shift dynamical system associated with a totally disconnected hyperbolic iterated function system (see Barnsley link). - Michel Marcus, Oct 06 2013

From Jean-Christophe Hervé, Oct 26 2014: (Start)

For n > 0, a(n) is also the number of orbits of size n of the transform associated to the Kolakoski sequence A000002 (and this is true for any map with 2^n periodic points of period n). The Kolakoski transform changes a sequence of 1's and 2's by the sequence of the lengths of its runs. The Kolakoski sequence is one of the two fixed points of this transform, the other being the same sequence without the initial term. A025142 and A025143 are the periodic points of the orbit of size 2. A027375(n) = n*a(n) gives the number of periodic points of minimal period n.

For n > 1, this sequence is equal to A059966 and to A060477, and for n = 1, a(1) = A059966(1)+1 = A060477(1)-1; this because the n-th term of all 3 sequences is equal to (1/n)*sum_{d|n} mu(n/d)*(2^d+e), with e = -1/0/1 for resp. A059966/this sequence/A060477, and sum_{d|n} mu(n/d) equals 1 for n = 1 and 0 for all n > 1. (End)

Warning: A000031 and A001037 are easily confused, since they have similar formulas.

From Petros Hadjicostas, Jul 14 2020: (Start)

Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.

Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) = A001221(N) being the number of distinct primes of N).

For N = 3, a = phi(3)/2 + omega(3) = 2/2 + 1 = 2 and b = omega(3) - 1 = 0. It follows that D(w, N=3) = A000079(w) = 2^w.

For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.

For N = 3, we get that c(w, N=3) = A000079(w)/w = 2^w/w.

He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)

Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).

For N = 3, we saw that a = 2 and b = 0, and hence d*(w, N=3) = a(w) = Sum_{k | w} (mu(k)/k) * 2^(w/k) / (w/k) = (1/w) * Sum_{k | w} mu(k) * 2^(w/k) for w >= 1. See Table 1 on p. 6 in Kam Cheong Au (2020). (End)

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).

In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011

Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019

(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021

From Jianing Song, Nov 13 2021: (Start)

a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.

a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).

a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.

T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.

From R. J. Mathar, Jul 31 2008: (Start)

This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:

This array begins as follows:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63

0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330

0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463

0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598

0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228

0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060

0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175

0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248

0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885

0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842

0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220

0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440

...

It is essentially symmetric: A(r,r+i) = A(r,r-i+1).

Some of the diagonals are:

A(r,r+1): A000108

A(r,r): A022553

A(r,r-1): A000108

A(r,r+2): A000150

A(r,r+3): A050181

A(r,r+4): A050182

A(r,r+5): A050183

A(r,r-2): A000150 (End)

Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019