A056391 -id:A056391 - cp's OEIS frontend

A056327 Number of reversible string structures with n beads using exactly three different colors.

0, 0, 1, 4, 15, 50, 160, 502, 1545, 4730, 14356, 43474, 131145, 395150, 1188580, 3572902, 10732065, 32225810, 96733636, 290322394, 871200825, 2614097750, 7843255300, 23531775502, 70599259185, 211805902490

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly three different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(4)=4, the color patterns are ABCA, ABBC, AABC, and ABAC. The first two are achiral. - _Robert A. Russell_, Oct 14 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36).

Column 3 of A284949.
Cf. A056310.
Cf. A000392 (oriented), A320526 (chiral), A304973 (achiral).

I:=[0,0,1,4,15,50,160]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

k=3; Table[(StirlingS2[n,k] + If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n,30}] (* Robert A. Russell, Oct 15 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k=3; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n,30}] (* Robert A. Russell, Oct 15 2018 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 1, 4, 15, 50, 160}, 30] (* Robert A. Russell, Oct 15 2018 *)

m=40; v=concat([0,0,1,4,15,50,160], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018

a(n) = A001998(n-1) - A005418(n).
G.f.: x^3*(3*x^4 - 8*x^3 + 3*x^2 + 2*x - 1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)). - Colin Barker, Sep 23 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=3 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
a(n) = (A000392(n) + A304973(n)) / 2 = A000392(n) - A320526(n) = A320526(n) + A304973(n). (End)

A056354 Number of bracelet structures using a maximum of four different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 33, 73, 237, 703, 2433, 8309, 30108, 108991, 403262, 1497070, 5607437, 21076571, 79595990, 301492045, 1145560579, 4363503684, 16660204452, 63741248201, 244339646708, 938255682551, 3608668388957, 13900021844558, 53614340398327, 207062143625711

Offset: 0

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A056292, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..4} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(26) and beyond from Andrew Howroyd, Oct 25 2019

A056355 Number of bracelet structures using a maximum of five different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 36, 89, 322, 1137, 4704, 19839, 88508, 399680, 1839947, 8533488, 39893901, 187393550, 884153396, 4185740195, 19876594537, 94633345608, 451615319433, 2159769331317, 10348546548695, 49672000435724, 238804871206358, 1149792978954373, 5543621482141513

Offset: 0

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A056293, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..5} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(25) and beyond from Andrew Howroyd, Oct 25 2019

A056375 Number of step shifted (decimated) sequences using a maximum of six different symbols.

Original entry on oeis.org

6, 36, 126, 756, 2016, 23976, 46956, 435456, 1683576, 15128856, 36284472, 547204896, 1088416056, 13060989936, 58782164616, 352913845536, 1057916846196, 16926689693376, 33853322280036, 457078896068256, 1828085963706576

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

G. C. Greubel, Table of n, a(n) for n = 1..1275
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Cf. A056414.

A row or column of A132191.

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[k, #] &], 0], {k, 1, n}]; Table[a[6, n], {n, 1, 21}] (* Jean-François Alcover, Dec 04 2015 *)

The cycle index is implicit in Titsworth.

Sequences A056372-A056375 fit a general formula, implemented in PARI/GP as follows: { a(m,n) = sum(k=1, n, if(gcd(k, n)==1, m^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0); ) / eulerphi(n) }. - Max Alekseyev, Nov 08 2007

More terms from Max Alekseyev, Nov 08 2007

A056412 Number of step cyclic shifted sequences using a maximum of four different symbols.

Original entry on oeis.org

4, 10, 20, 55, 76, 430, 460, 2605, 5164, 26962, 38572, 367645, 431780, 3203430, 8993804, 33860125, 63177820, 636462350, 803796700, 6886280971, 17456594380, 79965550558, 139069427020, 1466861706095, 2251803181492, 14434628481170, 37066691779180, 214483458079665, 354963555781060, 4803855154772166

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 4 of A285548.

Cf. A002729.

M[j_, L_] := Module[{m = 1}, While[Sum[j^i, {i, 0, m-1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[1/M[j, n / GCD[n, u*(j-1) + t]], {u, 0, n - 1}]; CB[n_, k_] = If[n==1, k, 1/(n*EulerPhi[n]) * Sum[ If[1 == GCD[n, j], k^c[j, t, n], 0] , {t, 0, n-1}, {j, 1, n-1}]]; Table[Print[cb = CB[n, 4]]; cb, {n, 1, 30}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,4),", "));
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A056455 Palindromes using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 24, 24, 240, 240, 1560, 1560, 8400, 8400, 40824, 40824, 186480, 186480, 818520, 818520, 3498000, 3498000, 14676024, 14676024, 60780720, 60780720, 249401880, 249401880, 1016542800, 1016542800, 4123173624, 4123173624, 16664094960, 16664094960

Offset: 1

Marks R. Nester

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-26,26,24,-24).

Cf. A056450, A000919.

k=4; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 4!*stirling((n+1)\2, 4, 2); \\ Altug Alkan, Sep 25 2018

a(n) = 4! * Stirling2( [(n+1)/2], 4).
G.f.: 24*x^7/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)). - Colin Barker, May 05 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=4 is the number of symbols. - Robert A. Russell, Sep 25 2018

A056456 Number of palindromes of length n using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 120, 120, 1800, 1800, 16800, 16800, 126000, 126000, 834120, 834120, 5103000, 5103000, 29607600, 29607600, 165528000, 165528000, 901020120, 901020120, 4809004200, 4809004200, 25292030400

Offset: 1

Marks R. Nester

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-71,71,154,-154,-120,120).

Cf. A056451, A001118.

k=5; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)
LinearRecurrence[{1, 14, -14, -71, 71, 154, -154, -120, 120}, {0, 0, 0, 0, 0, 0, 0, 0, 120}, 30] (* Vincenzo Librandi, Sep 29 2018 *)

a(n) = 5!*stirling((n+1)\2, 5, 2); \\ Altug Alkan, Sep 25 2018

a(n) = 5! * Stirling2( [(n+1)/2], 5).
G.f.: -120*x^9/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). - Colin Barker, Sep 03 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=5 is the number of symbols. - Robert A. Russell, Sep 25 2018

A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.

Original entry on oeis.org

4, 10, 16, 40, 64, 160, 256, 640, 1024, 2560, 4096, 10240, 16384, 40960, 65536, 163840, 262144, 655360, 1048576, 2621440, 4194304, 10485760, 16777216, 41943040, 67108864, 167772160, 268435456, 671088640, 1073741824, 2684354560, 4294967296, 10737418240

Offset: 1

Marks R. Nester

Old name was: "Number of periodic palindromes using a maximum of four different symbols".

Number of necklaces with n beads and 4 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

			G.f. = 4*x + 10*x^2 + 16*x^3 + 40*x^4 + 64*x^5 + 160*x^6 + 256*x^7 + 640*x^8 + ...
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A029744, A038754, A056450.

Column 4 of A284855.

[(9*2^n + (-2)^n)/4 : n in [1..50]]; // Wesley Ivan Hurt, Nov 24 2016

A056486:=n->(9*2^n + (-2)^n)/4: seq(A056486(n), n=1..50); # Wesley Ivan Hurt, Nov 24 2016

CoefficientList[Series[-1+(1+4*x+6*x^2)/(1-4*x^2),{x,0,30}],x] (* Herbert Kociemba, Nov 24 2016 *)
k=4; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, 1, 30}] (* Robert A. Russell, Sep 21 2018 *)

a(n) = (9*2^n+(-2)^n)/4; \\ Altug Alkan, Sep 21 2018

[2^(n-2)*(9+(-1)^n) for n in range(1,51)] # G. C. Greubel, Mar 23 2024

a(n) = 4^((n+1)/2) for n odd, a(n) = 4^(n/2)*5/2 for n even.
From Colin Barker, Jul 08 2012: (Start)
a(n) = 4*a(n-2).
G.f.: 2*x*(2+5*x)/((1-2*x)*(1+2*x)). (End)
G.f.: -1 + (1+4*x+6*x^2)/(1-4*x^2). - Herbert Kociemba, Nov 24 2016
E.g.f.: 5*sinh(x)^2 + 2*sinh(2*x). - Ilya Gutkovskiy, Nov 24 2016
a(n) = ( 4^floor((n+1)/2) + 4^ceiling((n+1)/2) )/2. - Robert A. Russell, Sep 21 2018

Better name from Ralf Stephan, Jul 18 2013

A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols.

Original entry on oeis.org

2, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

Also number of aperiodic necklaces (Lyndon words) with two colors that are the same when turned over.

			a(1) = 2 with aaa... and bbb..., a(2) = 1 with ababab..., a(3) = 2 with aabaab... and abbabb..., a(4) = 3 with aaabaaab... and aabbaabb... and abbbabbb.... - _Michael Somos_, Nov 29 2016

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for a pdf file of Chap. 2]

Index entries for sequences related to Lyndon words

Column 2 of A284856.

Cf. A029744, A056391, A056458.

mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)),{n,mx}]; CoefficientList[Series[gf[x,2],{x,0,mx}],x] (* Herbert Kociemba, Nov 29 2016 *)

Sum_{d|n} mu(d)*b(n/d), where b(n) = A029744(n+1). [Corrected by Petros Hadjicostas, Oct 15 2017. The original formula referred to a previous version of sequence A029744 that had a different offset.]
More generally, let gf(k) be the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors. Then gf(k): Sum_{n >= 1} mu(n)*Sum_{i=0..2} binomial(k,i)*x^(n*i)/(1 - k*x^(2*n)). - Herbert Kociemba, Nov 29 2016
G.f.: Sum_{n >= 1} mu(n)*x^n*(2 + 3*x^n)/(1 - 2*x^(2*n)). The g.f. by Herbet Kociemba above, with k = 2, becomes Sum_{n>=1} mu(n)*(x^n + 1)^2/(1 - 2*x^(2*n)). The two formulae differ by the "undetermined" constant Sum_{n >= 1} mu(n). - Petros Hadjicostas, Oct 15 2017

More terms and additional comments from Christian G. Bower, Jun 22 2000

A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0

Offset: 1

Andrew Howroyd, Apr 04 2017

A string and its reverse are considered to be equivalent.

			Table starts:
1   2    3     4      5      6       7       8 ...
0   1    3     6     10     15      21      28 ...
0   4   15    36     70    120     189     280 ...
0   7   39   126    310    645    1197    2044 ...
0  18  132   540   1620   3990    8568   16632 ...
0  29  357  2034   7790  23295   58779  131012 ...
0  70 1131  8316  39370 140610  412965 1050616 ...
0 126 3276 32760 195300 839790 2882376 8388576 ...
...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A045625, A056314, A056315, A056316, A056317.

Cf. A277504, A143324.

b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)

b(n,k) = (k^n + k^(ceil(n/2))) / 2;
a(n,k) = sumdiv(n,d, moebius(n/d) * b(d,k));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print(););

T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.

cp's OEIS Frontend

A056327 Number of reversible string structures with n beads using exactly three different colors.

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A056354 Number of bracelet structures using a maximum of four different colored beads.

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A056355 Number of bracelet structures using a maximum of five different colored beads.

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A056375 Number of step shifted (decimated) sequences using a maximum of six different symbols.

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A056412 Number of step cyclic shifted sequences using a maximum of four different symbols.

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A056455 Palindromes using exactly four different symbols.

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A056456 Number of palindromes of length n using exactly five different symbols.

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