A034691 -id:A034691 - cp's OEIS frontend

A358369 Euler transform of 2^floor(n/2), (A016116).

1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636

Offset: 0

Peter Luschny, Nov 17 2022

nonn

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;
u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;
b*u(n - 1) + c*u(n - 2) end; u end:
EulerTransform := proc(a) local b;
b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),
d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:
a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

from typing import Callable
from functools import cache
from sympy import divisors
def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:
    @cache
    def u(n: int) -> int:
        if n < 2:
            return [u0, u1][n]
        return b * u(n - 1) + c * u(n - 2)
    return u
def EulerTransform(a: Callable) -> Callable:
    @cache
    def b(n: int) -> int:
        if n == 0:
            return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)
            for j in range(1, n + 1))
        return s // n
    return b
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

A055375 Euler transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 14, 21, 14, 5, 7, 26, 48, 48, 26, 7, 11, 45, 103, 131, 103, 45, 11, 15, 75, 198, 312, 312, 198, 75, 15, 22, 120, 366, 674, 830, 674, 366, 120, 22, 30, 187, 637, 1359, 1961, 1961, 1359, 637, 187, 30, 42, 284, 1078, 2584, 4302, 5066, 4302, 2584, 1078, 284, 42

Offset: 0

Christian G. Bower, May 16 2000

Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - Franklin T. Adams-Watters, Jan 10 2007

			Triangle begins
   1;
   1,  1;
   2,  3,   2;
   3,  7,   7,   3;
   5, 14,  21,  14,   5;
   7, 26,  48,  48,  26,   7;
  11, 45, 103, 131, 103,  45, 11;
  15, 75, 198, 312, 312, 198, 75, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A034899.
Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
T(2n,n) gives A360626.
Cf. A007318, A034691, A209406, A360634.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 14 2023

nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - Franklin T. Adams-Watters, Jan 10 2007

Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - Alois P. Heinz, Dec 05 2023

A232580 Number of binary sequences of length n that contain at least one contiguous subsequence 011.

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 31, 74, 168, 369, 792, 1672, 3487, 7206, 14788, 30185, 61356, 124308, 251199, 506578, 1019920, 2050785, 4119280, 8267216, 16580799, 33236622, 66594636, 133385689, 267089188, 534692604, 1070217247, 2141780762, 4285739832, 8575004241

Offset: 0

Geoffrey Critzer, Nov 26 2013

From Gus Wiseman, Jun 26 2022: (Start)
Also the number of integer compositions of n + 1 with an even part other than the first or last. For example, the a(3) = 1 through a(5) = 12 compositions are:
  (121)  (122)   (123)
         (221)   (141)
         (1121)  (222)
         (1211)  (321)
                 (1122)
                 (1212)
                 (1221)
                 (2121)
                 (2211)
                 (11121)
                 (11211)
                 (12111)
The odd version is A274230.
(End)

			a(4) = 4 because we have: 0011, 0110, 0111, 1011.

Colin Barker, Table of n, a(n) for n = 0..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 34.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

The complement is counted by A000071(n) = A001911(n) + 1.
For the contiguous pattern (1,1) or (0,0) we have A000225.
For the contiguous pattern (1,0,1) or (0,1,0) we have A000253.
For the contiguous pattern (1,0) or (0,1) we have A000295.
Numbers whose binary expansion is of this type are A004750.
For the contiguous pattern (1,1,1) or (0,0,0) we have A050231.
The not necessarily contiguous version is A324172.
Cf. A034691, A261983, A274230, A335455, A335457, A335458, A335516.

nn=40;a=x/(1-x);CoefficientList[Series[a^2 x/(1-a x)/(1-2x),{x,0,nn}],x]
(* second program *)
Table[Length[Select[Tuples[{0,1},n],MatchQ[#,{_,0,1,1,_}]&]],{n,0,10}] (* Gus Wiseman, Jun 26 2022 *)

concat(vector(3), Vec(x^3/(-2*x^4+x^3+4*x^2-4*x+1) + O(x^40))) \\ Colin Barker, Nov 03 2016

O.g.f.: x^3/( (1-x)^2*(1-x^2/(1-x))*(1-2x) ).
a(n) ~ 2^n.
From Colin Barker, Nov 03 2016: (Start)
a(n) = (1 + 2^n - (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)).
a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4) for n > 3. (End)
a(n) = 2^n - Fibonacci(n+3) + 1. - Ehren Metcalfe, Dec 27 2018
E.g.f.: 2*exp(x/2)*(5*exp(x)*cosh(x/2) - 5*cosh(sqrt(5)*x/2) - 2*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 06 2022

A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 7, 2, 1, 0, 16, 16, 7, 2, 1, 0, 32, 42, 20, 7, 2, 1, 0, 64, 96, 54, 20, 7, 2, 1, 0, 128, 228, 140, 59, 20, 7, 2, 1, 0, 256, 512, 360, 156, 59, 20, 7, 2, 1, 0, 512, 1160, 888, 422, 162, 59, 20, 7, 2, 1, 0, 1024, 2560, 2168, 1088, 442, 162, 59, 20, 7, 2, 1

Offset: 0

M. F. Hasler, Jul 24 2017

T(n,k) is the number of multisets of exactly k binary words with a total of n letters and each word beginning with 1. T(4,2) = 7: {1,100}, {1,101}, {1,110}, {1,111}, {10,10}, {10,11}, {11,11}. - Alois P. Heinz, Sep 18 2017

			The triangle starts:
1;
0    1;
0    2    1;
0    4    2    1;
0    8    7    2    1;
0   16   16    7    2   1;
0   32   42   20    7   2   1;
0   64   96   54   20   7   2  1;
0  128  228  140   59  20   7  2  1;
0  256  512  360  156  59  20  7  2  1;
0  512 1160  888  422 162  59 20  7  2  1;
0 1024 2560 2168 1088 442 162 59 20  7  2  1;
(...)

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A034691 (row sums), A000007 (column k=0), A011782 (column k=1), A178945(n-1) (column k=2).
The reverse of the n-th row converges to A034899.
Cf. A000079, A209406, A292506.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(binomial(2^(i-1)+j-1, j)*
         b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 12 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[Binomial[2^(i - 1) + j - 1, j] b[n - i j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

G.f.: 1 / Product_{j>=1} (1-y*x^j)^(2^(j-1)). - Alois P. Heinz, Sep 18 2017

A298971 Number of compositions of n that are proper powers of Lyndon words.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 20, 9, 35, 1, 69, 1, 110, 21, 188, 1, 381, 7, 632, 59, 1184, 1, 2300, 1, 4115, 189, 7712, 25, 14939, 1, 27596, 633, 52517, 1, 101050, 1, 190748, 2247, 364724, 1, 703331, 19, 1342283, 7713, 2581430, 1, 4985609, 193

Offset: 1

Gus Wiseman, Jan 30 2018

nonn

a(n) is the number of compositions of n that are not Lyndon words but are of the form p * p * ... * p where * is concatenation and p is a Lyndon word.

			The a(12) = 16 compositions: 111111111111, 1111211112, 11131113, 112112112, 11221122, 114114, 12121212, 123123, 131313, 132132, 1515, 222222, 2424, 3333, 444, 66.

Cf. A000005, A000031, A000740, A000961, A001045, A008965, A019536, A034691, A051953, A052823, A059966, A060223, A178472, A185700, A296302, A296373.

Table[Sum[DivisorSum[d,MoebiusMu[d/#]*(2^#-1)&]/d,{d,Most@Divisors[n]}],{n,100}]

a(n) = sumdiv(n, d, (2^d-1)*(eulerphi(n/d)-moebius(n/d))/n); \\ Michel Marcus, Jan 31 2018

a(n) = Sum_{d|n} (2^d-1)*(phi(n/d)-mu(n/d))/n.

a(n) = A008965(n) - A059966(n).

A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 2, 6, 14, 40, 84, 224, 484, 1134, 2480, 5632, 12036, 26624, 56896, 122640, 261078, 557056, 1176876, 2490368, 5237360, 11008704, 23057408, 48234496, 100635144, 209714400, 436154368, 905962860, 1878931264, 3892314112, 8052800160, 16642998272, 34359209436

Offset: 1

Gus Wiseman, Feb 03 2018

nonn

For prime p, a(p) = 2^(p-2)*p. - Jon E. Schoenfield, Feb 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A001787, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299149, A299151.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, n*2^(n-2)-
       add(a(d)*a(n/d), d=divisors(n) minus {1, n})/2)
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 07 2018

nn=50;
sys=Table[2^(n-1)*n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]

A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 4, 11, 32, 84, 230, 597, 1567, 4020, 10286, 25994, 65387, 163065, 404617, 997687, 2448220, 5977334, 14530835, 35173496, 84814982, 203760809, 487845377, 1164191563, 2769721073, 6570218773, 15542642042, 36671354125, 86306246887, 202637312099, 474684979292, 1109539437382

Offset: 0

Ilya Gutkovskiy, Oct 01 2018

nonn

Convolution of A010815 and A034899.

Euler transform of A000225.

Alois P. Heinz, Table of n, a(n) for n = 0..3171
N. J. A. Sloane, Transforms

Cf. A000225, A007070, A010815, A034691, A034899, A319919.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
       d*(2^d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 13 2021

nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Exp[Sum[x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ A247003^2 * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A065446 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 15 2021

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
From Gus Wiseman, Apr 03 2025: (Start)
Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
         {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
                    {{1},{2},{2}}  {{1,2},{2,3}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{1,3},{2,4}}
                                   {{1,4},{2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{2},{3}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A330942, A331639.
For constant instead of strict blocks we have A034729.
Without equal sizes we have A116540 (normal set multipartitions).
Without strict blocks we have A317583.
For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
For equal sums instead of sizes we have A382429, non-strict blocks A326518.
Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

a(n) = Sum_{d|n} A330942(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

A334273 Numbers k such that the k-th composition in standard order is both a reversed necklace and a co-necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 141, 143, 146, 147

Offset: 1

Gus Wiseman, Apr 25 2020

nonn

A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklaces are defined similarly, except with greater instead of less.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of all reversed necklace co-necklaces begins:
    0: ()            31: (1,1,1,1,1)       69: (4,2,1)
    1: (1)           32: (6)               71: (4,1,1,1)
    2: (2)           33: (5,1)             73: (3,3,1)
    3: (1,1)         34: (4,2)             74: (3,2,2)
    4: (3)           35: (4,1,1)           75: (3,2,1,1)
    5: (2,1)         36: (3,3)             77: (3,1,2,1)
    7: (1,1,1)       37: (3,2,1)           79: (3,1,1,1,1)
    8: (4)           39: (3,1,1,1)         85: (2,2,2,1)
    9: (3,1)         42: (2,2,2)           87: (2,2,1,1,1)
   10: (2,2)         43: (2,2,1,1)         91: (2,1,2,1,1)
   11: (2,1,1)       45: (2,1,2,1)         95: (2,1,1,1,1,1)
   15: (1,1,1,1)     47: (2,1,1,1,1)      127: (1,1,1,1,1,1,1)
   16: (5)           63: (1,1,1,1,1,1)    128: (8)
   17: (4,1)         64: (7)              129: (7,1)
   18: (3,2)         65: (6,1)            130: (6,2)
   19: (3,1,1)       66: (5,2)            131: (6,1,1)
   21: (2,2,1)       67: (5,1,1)          132: (5,3)
   23: (2,1,1,1)     68: (4,3)            133: (5,2,1)

The aperiodic case is A334266.
Compositions of this type are counted by A334271.
Normal sequences of this type are counted by A334272.
Another ranking of the same compositions is A334274 (binary expansion).
Binary (or reversed binary) necklaces are counted by A000031.
Necklace compositions are counted by A008965.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Reversed necklaces are A333943.
- Co-necklaces are A333764.
- Reversed co-necklaces are A328595.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
Cf. A019536, A034691, A059966, A060223, A329138, A334269, A334270.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,100],neckQ[Reverse[stc[#]]]&&coneckQ[stc[#]]&]

cp's OEIS Frontend

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