A008965 -id:A008965 - cp's OEIS frontend

A059076 Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged.

0, 0, 0, 0, 0, 0, 1, 2, 6, 14, 30, 62, 128, 252, 495, 968, 1866, 3600, 6917, 13286, 25476, 48916, 93837, 180314, 346554, 666996, 1284570, 2477342, 4781502, 9240012, 17871708, 34604066, 67060746, 130085052, 252548760, 490722344

Offset: 0

Henry Bottomley, Dec 22 2000

nonn

Number of chiral bracelets with n beads and two colors.

			For n=6, the only chiral pair is AABABB-AABBAB.  For n=7, the two chiral pairs are AAABABB-AAABBAB and AABABBB-AABBBAB. - _Robert A. Russell_, Sep 24 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Efficient Construction of Long Orientable Sequences, arXiv:2401.14341 [cs.DS], 2024.
Petros Hadjicostas, Formulas for chiral bracelets, 2019; see Section 5.
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012

Column 2 of A293496.
Cf. A059053.
Column 2 of A305541.
Equals (A000031 - A164090) / 2.
a(n) = (A052823(n) - A027383(n-2)) / 2.

nn=35;Table[CoefficientList[Series[CycleIndex[CyclicGroup[n],s]-CycleIndex[DihedralGroup[n],s]/.Table[s[i]->2,{i,1,n}],{x,0,nn}],x],{n,1,nn}]//Flatten  (* Geoffrey Critzer, Mar 26 2013 *)
mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-2*x^n]/n, {n, mx}]-(1+x)^2/(1-2*x^2))/2, {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
terms = 36; a29[0] = 1; a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#) & ]/(2*n); Array[a29, 36, 0] - LinearRecurrence[{0, 2}, {1, 2, 3}, 36] (* Jean-François Alcover, Nov 05 2017 *)
k = 2; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n)(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 0] (* Robert A. Russell, Sep 24 2018 *)

a(n) = A000031(n) - A000029(n) = A000029(n) - A029744(n) = (A000031(n) - A029744(n))/2 = A008965(n) - A091696(n)
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n - (1 + x)^2/(1 - 2*x^2))/2. - Herbert Kociemba, Nov 02 2016
For n > 0, a(n) = -(k^floor((n + 1)/2) + k^ceiling((n + 1)/2))/4 + (1/(2*n))* Sum_{d|n} phi(d)*k^(n/d), where k = 2 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

A294859 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in lexicographic order.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 1, 2, 1, 3, 4, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 4, 1, 2, 3, 1, 3, 2, 1, 5, 2, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1

Offset: 1

Gus Wiseman, Dec 18 2017

			Triangle of Lyndon compositions begins:
(1),
(2),
(12),(3),
(112),(13),(4),
(1112),(113),(122),(14),(23),(5),
(11112),(1113),(1122),(114),(123),(132),(15),(24),(6),
(111112),(11113),(11122),(1114),(11212),(1123),(1132),(115),(1213),(1222),(124),(133),(142),(16),(223),(25),(34),(7).

Cf. A000740, A001037, A001045, A008965, A059966, A060223, A066099, A101211, A102659, A124734, A185700, A228369, A281013, A296302, A296373, A296656.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ],OrderedQ[PadRight[{#1,#2}]]&],{n,7}]

Row n is a concatenation of A059966(n) Lyndon words with total length A000740(n).

A323859 Number of binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 6, 8, 19, 16, 56, 40, 152, 184, 432, 376, 2132, 1264, 4728, 8768, 20688, 15424, 87656, 55192, 315128, 399520, 762984, 729448, 5595408, 4026576, 10325712, 19884504, 57527804, 37025584, 286340544, 138547336, 805335364, 1041204704, 2021176512, 3926827328

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A000031.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(4) = 19 binary toroidal necklaces:
  [0 0 0 0] [0 0 0 1] [0 0 1 1] [0 1 0 1] [0 1 1 1] [1 1 1 1]
.
  [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
  [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
.
  [0] [0] [0] [0] [0] [1]
  [0] [0] [0] [1] [1] [1]
  [0] [0] [1] [0] [1] [1]
  [0] [1] [1] [1] [1] [1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A008965, A179043, A184271, A323351.

Cf. A323858, A323859, A323865, A323870.

matcyc[m_]:=Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[If[n==0,1,Length[Union[First/@matcyc/@Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n])]]],{n,0,10}]

U(n, m, k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * k^(n*m/lcm(c, d))))
a(n) = if(n<1, n==0, sumdiv(n, d, U(n/d, d, 2))) \\ Andrew Howroyd, Jan 24 2023

a(n) = (1/n) * Sum_{d|n} Sum_{e|d, f|(n/d)} phi(e) * phi(f) * 2^(n/lcm(d,n/d)). [Ethier]

A325547 Number of compositions of n with strictly increasing differences.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 18, 24, 30, 45, 57, 71, 96, 120, 148, 192, 235, 286, 354, 431, 518, 628, 752, 893, 1063, 1262, 1482, 1744, 2046, 2386, 2775, 3231, 3733, 4305, 4977, 5715, 6536, 7507, 8559, 9735, 11112, 12608, 14252, 16177, 18265, 20553, 23204, 26090, 29223

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 11 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)
       (11)  (12)  (13)   (14)   (15)
             (21)  (22)   (23)   (24)
                   (31)   (32)   (33)
                   (112)  (41)   (42)
                   (211)  (113)  (51)
                          (212)  (114)
                          (311)  (213)
                                 (312)
                                 (411)
                                 (2112)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A179269 (breakdown by width)
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 27 2019

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 27 2019

A325548 Number of compositions of n with strictly decreasing differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)
       (11)  (12)  (13)   (14)   (15)    (16)   (17)
             (21)  (22)   (23)   (24)    (25)   (26)
                   (31)   (32)   (33)    (34)   (35)
                   (121)  (41)   (42)    (43)   (44)
                          (122)  (51)    (52)   (53)
                          (131)  (132)   (61)   (62)
                          (221)  (141)   (133)  (71)
                                 (231)   (142)  (134)
                                 (1221)  (151)  (143)
                                         (232)  (152)
                                         (241)  (161)
                                         (331)  (233)
                                                (242)
                                                (251)
                                                (332)
                                                (341)
                                                (431)
                                                (1331)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
       j-l b(n, 0$2):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 27 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,15}]

a(26)-a(44) from Lars Blomberg, May 30 2019

a(45)-a(52) from Alois P. Heinz, Jan 27 2024

A328598 Number of compositions of n with no part circularly followed by a divisor.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 7, 12, 11, 22, 26, 55, 63, 99, 149, 215, 324, 458, 699, 1006, 1492, 2185, 3202, 4734, 6928, 10242, 14951, 22023, 32365, 47557, 69905, 102633, 150983, 221712, 325918, 478841, 703647, 1034103, 1519431, 2233061, 3281003, 4821790, 7085358

Offset: 0

Gus Wiseman, Oct 24 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Circularity means the last part is followed by the first.

			The a(5) = 2 through a(12) = 22 compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)
  (3,2)  (3,4)  (5,3)  (4,5)    (4,6)      (3,8)    (7,5)
         (4,3)         (5,4)    (6,4)      (4,7)    (2,3,7)
         (5,2)         (7,2)    (7,3)      (5,6)    (2,7,3)
                       (2,4,3)  (2,3,5)    (6,5)    (3,2,7)
                       (3,2,4)  (2,5,3)    (7,4)    (3,4,5)
                       (4,3,2)  (3,2,5)    (8,3)    (3,5,4)
                                (3,5,2)    (9,2)    (3,7,2)
                                (5,2,3)    (2,4,5)  (4,3,5)
                                (5,3,2)    (4,5,2)  (4,5,3)
                                (2,3,2,3)  (5,2,4)  (5,3,4)
                                (3,2,3,2)           (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)

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

The necklace version is A328600, or A318729 without singletons.
The version with singletons is A318726.
The non-circular version is A328460.
Also forbidding parts circularly followed by a multiple gives A328599.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A178470, A318748, A328187, A328508, A328593, A328597, A328601, A328603, A328609.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,0,10}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i,j)->i%j<>0)))} \\ Andrew Howroyd, Oct 26 2019

a(n > 0) = A318726(n) - 1.

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 224, 226, 228, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 255, 256, 384, 392, 400, 416, 420, 424, 426, 448, 450

Offset: 0

Antti Karttunen, May 16 2015

These correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of binary necklaces. See the last example.
Indexing starts from zero, because a(0) = 0 is a special case.
If k is a member then so also is 2*k, i.e., k with 0 appended to the end of its binary representation.
If k is a member then so also is A004755(k), i.e., k with 1 prepended to the front of its binary representation.
One obtains A065609 if one erases the most significant bit of each term [as A053645(a(n))] and then discards any zero-terms produced from the terms that originally were powers of two (A000079).
First differs from A328607 in lacking 108, with binary expansion 1101100. If we define a dual-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is a dual-necklace. - Gus Wiseman, Nov 04 2019

			For n = 5, with binary representation "101", if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction, we get "110" = 6 > 5, so 5 can be made larger by such rotations, and thus is NOT included in this sequence.
For n = 6, with binary representation "110", no such rotation will yield a larger number, and thus 6 is included in this sequence.
For n = 28, with binary representation "11100", if we rotate non-msb bits towards right, we get additional numbers 22, 19 and 25 (with binary representations "10110", "10011", "11001") before coming to 28 again, and 28 is the largest of these numbers, thus 28 is included in this sequence.
  Also, if we discard the most significant bit of each and consider them just as binary strings, then A053645(28) = 12 is the lexicographically largest representative of {"1100", "0110", "0011", "1001"}, which is the complete set of representatives for a particular equivalence class of binary necklaces, obtained by rotating all bits of binary string "1100" successively towards right or left.

Antti Karttunen, Table of n, a(n) for n = 0..16637
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Complement: A257739.
Odd terms: A000225.
Subsequence of A065609.
Cf. A004755, A053645, A080541, A080542, A256999.
Subsequence: A258003.
The non-dual version is A328668.
The version involving all digits is A065609.
The non-dual reversed version is A328607.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A069010, A275692, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,110],#<=1||reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 04 2019 *)

A318731 Number of relatively prime Lyndon compositions (aperiodic necklaces of positive integers) with sum n.

Original entry on oeis.org

1, 0, 1, 2, 5, 7, 17, 27, 54, 93, 185, 324, 629, 1143, 2175, 4050, 7709, 14469, 27593, 52276, 99839, 190371, 364721, 698508, 1342170, 2580165, 4970952, 9585232, 18512789, 35787985, 69273665, 134211600, 260300799, 505278705, 981706783

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(6) = 7 relatively prime Lyndon compositions are 15, 114, 132, 123, 1113, 1122, 11112.
The a(7) = 17 relatively prime Lyndon compositions:
  16, 25, 34,
  115, 142, 124, 133, 223,
  1114, 1213, 1132, 1123, 1222,
  11113, 11212, 11122,
  111112.

Cf. A000740, A000837, A008965, A059966, A100953, A296302.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ[#]&&GCD@@#==1&]],{n,10}]

Moebius transform of A059966. Second Moebius transform of A008965.

A318748 Number of integer compositions of n that have only one part or whose consecutive parts are coprime and the last and first part are also coprime.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 43, 82, 151, 285, 535, 1005, 1883, 3533, 6631, 12460, 23407, 43952, 82538, 154999, 291088, 546674, 1026687, 1928118, 3621017, 6800300, 12771086, 23984329, 45042959, 84591339, 158863807, 298348613, 560303342, 1052258402, 1976157510

Offset: 0

Gus Wiseman, Sep 02 2018

nonn

			The a(5) = 13 compositions with adjacent parts coprime:
  (5)
  (41) (14) (32) (23)
  (311) (131) (113)
  (2111) (1211) (1121) (1112)
  (11111)
Missing from this list are (221), (212), and (122).

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

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A318728, A318745, A328609.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], vector(n, i, i > 1) + sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

a(n) = A328609(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

a(21)-a(35) from Alois P. Heinz, Sep 02 2018

Name corrected by Gus Wiseman, Nov 04 2019

A323871 Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 8, 53, 216, 3112, 13512, 272844, 2362412, 40898808, 295024104, 14045779864, 81055130520, 3040383692328, 61408850927280, 1661142087743940, 15337737297545400, 1128511554416582908, 9768588138876674856, 803306338873264137240, 15452347618762680730384

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (Lyndon word) case is A060223.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(3) = 8 aperiodic toroidal necklaces:
  [1 2 3] [1 3 2] [1 2 2] [1 1 2]
.
  [1] [1] [1] [1]
  [2] [3] [2] [1]
  [3] [2] [2] [2]

Andrew Howroyd, Table of n, a(n) for n = 1..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000670, A000740, A008965, A060223.

Cf. A323858, A323859, A323860, A323861, A323866, A323867, A323868, A323870.

List([1..30], A323871); # See A323861 for code; Andrew Howroyd, Aug 21 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[nrmmats[n],neckmatQ[#]&&apermatQ[#]&]],{n,6}]

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

cp's OEIS Frontend

A059076 Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged.

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A294859 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in lexicographic order.

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A323859 Number of binary toroidal necklaces of size n.

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PARI

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A325547 Number of compositions of n with strictly increasing differences.

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A325548 Number of compositions of n with strictly decreasing differences.

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Maple

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A328598 Number of compositions of n with no part circularly followed by a divisor.

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A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

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