cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A208777 T(n,k) is the number of n-bead necklaces labeled with numbers 1..k not allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 6, 1, 6, 9, 10, 12, 8, 1, 7, 11, 13, 18, 19, 14, 1, 8, 13, 16, 24, 30, 39, 20, 1, 9, 15, 19, 30, 41, 65, 71, 36, 1, 10, 17, 22, 36, 52, 91, 128, 152, 60, 1, 11, 19, 25, 42, 63, 117, 185, 293, 315, 108, 1, 12, 21, 28, 48, 74, 143, 242, 435, 658, 685
Offset: 1

Views

Author

R. H. Hardin, Mar 01 2012

Keywords

Comments

Table starts
.1..2...3...4...5...6...7...8....9...10...11...12...13...14...15...16...17...18
.1..3...5...7...9..11..13..15...17...19...21...23...25...27...29...31...33...35
.1..4...7..10..13..16..19..22...25...28...31...34...37...40...43...46...49...52
.1..6..12..18..24..30..36..42...48...54...60...66...72...78...84...90...96..102
.1..8..19..30..41..52..63..74...85...96..107..118..129..140..151..162..173..184
.1.14..39..65..91.117.143.169..195..221..247..273..299..325..351..377..403..429
.1.20..71.128.185.242.299.356..413..470..527..584..641..698..755..812..869..926
.1.36.152.293.435.577.719.861.1003.1145.1287.1429.1571.1713.1855.1997.2139.2281
The transposed array (starting with index 0) appears as Table 2 in the Knopfmacher et al. reference. [Joerg Arndt, Aug 08 2012]

Examples

			All solutions for n=4, k=3:
..2....1....2....1....2....2....2....1....3....1....1....1
..3....2....2....2....2....3....2....1....3....1....2....1
..2....2....3....1....2....3....2....1....3....2....3....1
..3....2....3....2....2....3....3....2....3....2....2....1

Links

Crossrefs

Column 2 is A000031, col. 3 is A208772, col. 4 is A208773, col. 5 is A208774, col. 6 is A208775, col. 7 is A208776.

Programs

  • Mathematica
    T[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1+2*Cos[i*Pi/(k+1)])^(n/#)&], {i, 1, k}] // FullSimplify; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 05 2015, adapted from PARI *)
  • PARI
    /* from the Knopfmacher et al. reference */
default(realprecision,99); /* using floats */
sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
T(n,k)=sn(n,k);
matrix(22,22,n,k, round(T(n,k)) ) /* as matrix shown in comments */
/* Joerg Arndt, Aug 09 2012 */

A215335 Cyclically smooth Lyndon words with 3 colors.

Original entry on oeis.org

3, 2, 4, 7, 16, 30, 68, 140, 308, 664, 1476, 3248, 7280, 16286, 36768, 83160, 189120, 431046, 986244, 2261616, 5200776, 11984382, 27676612, 64031520, 148406224, 344500520, 800902564, 1864486560, 4346071600, 10142581552, 23696518916, 55420651440, 129742921992, 304014466080, 712985901856, 1673486122000
Offset: 1

Views

Author

Joerg Arndt, Aug 13 2012

Keywords

Comments

We call a Lyndon word (x[1],x[2],...,x[n]) smooth if abs(x[k]-x[k-1]) <= 1 for 2<=k<=n, and cyclically smooth if abs(x[1]-x[n]) <= 1.

Examples

			The cyclically smooth necklaces (N) and Lyndon words (L) of length 4 with 3 colors (using symbols ".", "1", and "2") are:
    ....   1       .  N
    ...1   4    ...1  N L
    ..11   4    ..11  N L
    .1.1   2      .1  N
    .111   4    .111  N L
    .121   4    .121  N L
    1111   1       1  N
    1112   4    1112  N L
    1122   4    1122  N L
    1212   2      12  N
    1222   4    1222  N L
    2222   1       2  N
There are 12 necklaces (so A208772(4)=12) and a(4)=7 Lyndon words.

Links

Crossrefs

Cf. A208772 (cyclically smooth necklaces, 3 colors).
Cf. A215327 (smooth necklaces, 3 colors), A215328 (smooth Lyndon words, 3 colors).

Programs

  • Mathematica
    terms = 40;
sn[n_, k_] := 1/n Sum[EulerPhi[j] (1+2Cos[i Pi/(k+1)])^(n/j), {i, 1, k}, {j, Divisors[n]}];
vn = Table[Round[sn[n, 3]], {n, terms}];
vl = Table[Sum[MoebiusMu[n/d] vn[[d]], {d, Divisors[n]}], {n, terms}] (* Jean-François Alcover, Jul 22 2018, after Joerg Arndt *)
  • PARI
    default(realprecision,99); /* using floats */
sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
vn=vector(66,n, round(sn(n,3)) ); /* necklaces */
/* Lyndon words, via Moebius inversion: */
vl=vector(#vn,n, sumdiv(n,d, moebius(n/d)*vn[d]))

Formula

a(n) = sum_{ d divides n } moebius(n/d) * A208772(d).

A208716 Number of n-bead necklaces labeled with numbers 1..3 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

3, 5, 7, 12, 18, 34, 56, 111, 207, 427, 859, 1851, 3930, 8672, 19092, 42845, 96243, 218567, 497183, 1138084, 2610226, 6009662, 13861968, 32057868, 74260243, 172351415, 400589343, 932486879, 2173368730, 5071877864, 11849063220, 27711739481
Offset: 1

Views

Author

R. H. Hardin, Mar 01 2012

Keywords

Examples

			All solutions for n=4:
..2....1....1....2....2....1....2....1....1....1....3....2
..2....2....1....3....3....1....2....2....2....1....3....2
..2....3....1....3....2....2....3....1....2....1....3....2
..3....2....2....3....3....2....3....2....2....1....3....2

Links

Crossrefs

Column 3 of A208721.
Cf. A208772, A078057.

Formula

a(2n+1) = (1/2) * (A208772(2n+1) + A078057(n+1)). - Andrew Howroyd, Mar 03 2017
a(2n) = (1/2) * A208772(2n) + (1/4) * (A078057(n) + A078057(n+1)). - Andrew Howroyd, Mar 03 2017
Showing 1-3 of 3 results.

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