A305750 -id:A305750 - cp's OEIS frontend

A056323 Number of reversible string structures with n beads using a maximum of four different colors.

1, 1, 2, 4, 11, 31, 107, 379, 1451, 5611, 22187, 87979, 350891, 1400491, 5597867, 22379179, 89500331, 357952171, 1431743147, 5726775979, 22906841771, 91626580651, 366505274027, 1466017950379, 5864067607211

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.
Number of set partitions of an unoriented row of n elements with four or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the formula. - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 02 2022: (Start)
a(n) is the number of (unlabeled) 4-paths with n+6 vertices. (A 4-path with order n at least 6 can be constructed from a 5-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to an existing 4-clique containing an existing 4-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

			For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD. The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

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]

Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Index entries for linear recurrences with constant coefficients, signature (5,0,-20,16).

Cf. A032121.
Column 4 of A320750.
Cf. A124303 (oriented), A320934 (chiral), A305750 (achiral).
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

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]] (* A304972 *)
k=4; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,0,k}]/2,{n,0,40}] (* Robert A. Russell, Oct 28 2018 *)
LinearRecurrence[{5, 0, -20, 16}, {1, 1, 2, 4, 11}, 40] (* Robert A. Russell, Oct 28 2018 *)

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
For n > 0, a(n) = (16 + (-2)^n + 15*2^n + 4^n)/48. - Colin Barker, Nov 24 2012
G.f.: (1 - 4x - 3x^2 + 14x^3 - 5x^4) / ((1-x)*(1-4x)*(1-4x^2)). - Colin Barker, Nov 24 2012 [Adapted to offset 0 by Robert A. Russell, Nov 09 2018]
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A124303(n) + A305750(n)) / 2.
a(n) = A124303(n) - A320934(n) = A320934(n) + A305750(n).
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n). (End)

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

A027615 Number of 1's when n is written in base -2.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2

Offset: 0

Pontus von Brömssen, Nov 14 1997

Base -2 is also called "negabinary".
From Jianing Song, Oct 18 2018: (Start)
Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(n), n >= 0. See A320642 for the other half of f.
For k > 0, the earliest occurrence of k is n = A305750(k).
Conjecture: a(n) != A053737(n) if and only if there exists even k >= 4 such that n mod 2^k >= (5*2^(k+1) + 2)/3. If this holds, then the probability of a random chosen number n to satisfy a(n) != A053737(n) is 1/6. (End)

			A039724(7) = 11011 which has four 1's, so a(7) = 4.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 164.

Jianing Song, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Negative base.

Cf. A000120, A005351, A039724, A053737, A072894, A305750, A320642.

a[n_] := a[n] = a[Quotient[n - 1, -2]] + Mod[n, 2]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 23 2023 *)

a(n) = if(n==0, 0, a(n\(-2))+n%2) /* Jianing Song, Oct 18 2018 */

a(n) = 3*A072894(n+1) - 2*n - 3. Proof by Nikolaus Meyberg, following a conjecture by Ralf Stephan. - R. J. Mathar, Jan 11 2013
a(n) == n (mod 3). - Jianing Song, Oct 18 2018
a(n) = A000120(A005351(n)). - Michel Marcus, Oct 23 2018

A305749 T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 18, 8, 1, 1, 2, 3, 7, 12, 27, 27, 16, 1, 1, 2, 3, 7, 12, 30, 43, 54, 16, 1, 1, 2, 3, 7, 12, 31, 55, 107, 81, 32, 1, 1, 2, 3, 7, 12, 31, 58, 141, 171, 162, 32, 1, 1, 2, 3, 7, 12, 31, 59, 159, 266, 427, 243, 64, 1, 1, 2, 3, 7, 12, 31, 59, 163, 312, 688, 683, 486, 64, 1

Offset: 1

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABC are equivalent, as are AAABB and BBBAA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a loop are equivalent, so for loops AAABCB = BAAABC = CBAAAB.

			The array begins at T(1,1):
1  1   1    1    1     1     1     1     1     1     1     1     1 ...
1  2   2    2    2     2     2     2     2     2     2     2     2 ...
1  2   3    3    3     3     3     3     3     3     3     3     3 ...
1  4   6    7    7     7     7     7     7     7     7     7     7 ...
1  4   9   11   12    12    12    12    12    12    12    12    12 ...
1  8  18   27   30    31    31    31    31    31    31    31    31 ...
1  8  27   43   55    58    59    59    59    59    59    59    59 ...
1 16  54  107  141   159   163   164   164   164   164   164   164 ...
1 16  81  171  266   312   334   338   339   339   339   339   339 ...
1 32 162  427  688   883   963   993   998   999   999   999   999 ...
1 32 243  683 1313  1774  2069  2169  2204  2209  2210  2210  2210 ...
1 64 486 1707 3407  5103  6119  6634  6789  6834  6840  6841  6841 ...
1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
a(n) are the terms of this array read by antidiagonals.
For T(4,3)=6, the achiral pattern rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. The achiral pattern loops are AAAA, AAAB, AABB, ABAB, AABC, and ABAC.

Columns 1-6 are A057427, A016116, A182522, A305750, A305751, and A305752.

Columns converge to the right to A080107.

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]]; (* A304972 *)
Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten

T(n,k) = Sum_{j=0..k} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0 <= n <= 1 & n==k].

T(n,k) = Sum_{j=1..k} A304972(n,j).

A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 30, 130, 532, 2006, 7626, 28401, 106260, 396435, 1486147, 5580130, 21032880, 79486763, 301317282, 1145123672, 4362804633, 16658456825, 63738451998, 244332656201, 938244497740, 3608640426930, 13899977105315, 53614228550220, 207061964668740, 800639722002163, 3099251007215286, 12009598156277090, 46582685655751645, 180850428684482360

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056292 and A305750, which can be used in conjunction with the first formula.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 4 of A320742.

Cf. A056292 (oriented), A056354 (unoriented), A305750 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
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]] (* A304972 *)
k=4; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=4 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n).

A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Column 4 of A320751.

Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral).

Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 40] (* or *)
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]] (* A304972 *)
k=4; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

a(n) = (A124303(n) - A305750(n))/2.
a(n) = A124303(n) - A056323(n).
a(n) = A056323(n) - A305750(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(2*m) = (16^m - 4*4^m)/48.
a(2*m-1) = (16^m - 4*4^m)/192.
a(n) = (4^n - 4^floor(n/2+1))/48.
G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018
a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018

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A056323 Number of reversible string structures with n beads using a maximum of four different colors.

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A027615 Number of 1's when n is written in base -2.

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A305749 T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).

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A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

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A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

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