A215327
Smooth necklaces with 3 colors.
Original entry on oeis.org
1, 3, 5, 8, 15, 27, 58, 115, 252, 541, 1196, 2629, 5894, 13156, 29667, 66978, 151966, 345497, 788396, 1802678, 4133161, 9495317, 21861393, 50423468, 116514553, 269666605, 625108573, 1451128479, 3373267275, 7851415838, 18296568717
Offset: 0
The smooth pre-necklaces, 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
..1. 3 .1.
..11 4 ..11 N L
..12 4 ..12 N L
.1.1 2 .1 N
.11. 3 11.
.111 4 .111 N L
.112 4 .112 N L
.121 4 .121 N L
.122 4 .122 N L
1111 1 1 N
1112 4 1112 N L
1121 3 121
1122 4 1122 N L
1212 2 12 N
1221 3 221
1222 4 1222 N L
2222 1 2 N
There are 19 pre-necklaces, 15 necklaces, and 10 Lyndon words.
So a(4) = 15.
Cf.
A001867 (necklaces, 3 colors),
A215328 (smooth Lyndon words, 3 colors).
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
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.
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.
- Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.
Cf.
A208772 (cyclically smooth necklaces, 3 colors).
Cf.
A215327 (smooth necklaces, 3 colors),
A215328 (smooth Lyndon words, 3 colors).
-
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 *)
-
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]))
A215330
Smooth Lyndon words with 4 colors.
Original entry on oeis.org
1, 4, 3, 8, 18, 47, 108, 268, 638, 1553, 3761, 9189, 22453, 55185, 135894, 335906, 832312, 2068066, 5149845, 12852750, 32138353, 80509495, 202013368, 507669048, 1277586867, 3219366610, 8122275225
Offset: 0
Cf.
A215329 (smooth necklaces, 4 colors),
A215328 (smooth Lyndon words, 3 colors).
A215331
Smooth necklaces with 5 colors.
Original entry on oeis.org
1, 5, 9, 16, 35, 76, 190, 455, 1156, 2911, 7438, 18992, 48902, 125968, 325975, 845202, 2197690, 5725854, 14951308, 39110371, 102490649, 269002564, 707096093, 1861183847, 4905172383, 12942843424
Offset: 0
The smooth pre-necklaces, necklaces (N), and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", "3", and "4") are:
.... 1 . N
...1 4 ...1 N L
..1. 3 .1.
..11 4 ..11 N L
..12 4 ..12 N L
.1.1 2 .1 N
.11. 3 11.
.111 4 .111 N L
.112 4 .112 N L
.121 4 .121 N L
.122 4 .122 N L
.123 4 .123 N L
1111 1 1 N
1112 4 1112 N L
1121 3 121
1122 4 1122 N L
1123 4 1123 N L
1212 2 12 N
1221 3 221
1222 4 1222 N L
1223 4 1223 N L
1232 4 1232 N L
1233 4 1233 N L
1234 4 1234 N L
2222 1 2 N
2223 4 2223 N L
2232 3 232
2233 4 2233 N L
2234 4 2234 N L
2323 2 23 N
2332 3 332
2333 4 2333 N L
2334 4 2334 N L
2343 4 2343 N L
2344 4 2344 N L
3333 1 3 N
3334 4 3334 N L
3343 3 343
3344 4 3344 N L
3434 2 34 N
3443 3 443
3444 4 3444 N L
4444 1 4 N
There are 43 pre-necklaces, 35 necklaces, and 26 Lyndon words.
So a(4) = 35.
Cf.
A215327 (smooth necklaces, 3 colors)
A215328 (smooth Lyndon words, 3 colors).
A215334
Smooth Lyndon words with 7 colors.
Original entry on oeis.org
1, 7, 6, 17, 42, 119, 305, 829, 2196, 5907, 15863, 42842, 115845, 314368, 854647, 2329087, 6358855, 17393327, 47652117, 130752969, 359270784, 988458115, 2722792878
Offset: 0
Cf.
A215328 (smooth Lyndon words, 3 colors),
A215327 (smooth necklaces, 3 colors).
Showing 1-5 of 5 results.
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