A135342 -id:A135342 - cp's OEIS frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

Original entry on oeis.org

1, 2, 3, 6, 9, 16, 27, 46, 77, 136, 239, 426, 769, 1400, 2571, 4762, 8857, 16568, 31139, 58734, 111165, 211044, 401695, 766418, 1465489, 2807672, 5388783, 10359850, 19946833, 38459624, 74251095, 143524762, 277742489, 538043664, 1043333935, 2025040766, 3933915349

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(0) = 1 through a(5) = 16 subsets:
  {}  {}   {}   {}       {}       {}
      {1}  {1}  {1}      {1}      {1}
           {2}  {2}      {2}      {2}
                {3}      {3}      {3}
                {1,3}    {4}      {4}
                {1,2,3}  {1,3}    {5}
                         {2,4}    {1,3}
                         {1,2,3}  {1,5}
                         {2,3,4}  {2,4}
                                  {3,5}
                                  {1,2,3}
                                  {1,3,5}
                                  {2,3,4}
                                  {3,4,5}
                                  {1,2,4,5}
                                  {1,2,3,4,5}

If the subset is required to contain n, we get A063776.

Cf. A000016, A051293, A065795, A082550, A135342, A327474, A327481.

with(numtheory):
b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:
a:= proc(n) option remember; `if`(n=0, 1, b(n)-1+a(n-1)) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Jan 13 2024

Table[Length[Select[Subsets[Range[n]],#=={}||IntegerQ[Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327475(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))-n+1 # Chai Wah Wu, Feb 22 2023

a(n) = A051293(n) + 1.

A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 19, 9, 3, 1, 1, 3, 9, 25, 25, 9, 3, 1, 1, 3, 9, 29, 51, 29, 9, 3, 1, 1, 3, 9, 31, 75, 75, 31, 9, 3, 1, 1, 3, 9, 31, 93, 151, 93, 31, 9, 3, 1, 1, 3, 9, 31, 105, 235, 235, 105, 31, 9, 3, 1

Offset: 1

Gus Wiseman, Sep 13 2019

All terms are odd.

			Triangle begins:
                         1
                       1   1
                     1   3   1
                   1   3   3   1
                 1   3   7   3   1
               1   3   9   9   3   1
             1   3   9  19   9   3   1
           1   3   9  25  25   9   3   1
         1   3   9  29  51  29   9   3   1
       1   3   9  31  75  75  31   9   3   1
     1   3   9  31  93 151  93  31   9   3   1
   1   3   9  31 105 235 235 105  31   9   3   1
The subsets counted in row n = 5:
  {1}  {2}      {3}          {4}      {5}
       {1,3}    {1,5}        {3,5}
       {1,2,3}  {2,4}        {3,4,5}
                {1,3,5}
                {2,3,4}
                {1,2,4,5}
                {1,2,3,4,5}

Row sums are A051293.
The sequence of rows converges to A066571.
The version for partitions is A327482.
Cf. A000016, A063776, A065795, A135342, A327474, A327475, A327483, A327484.

Table[Length[Select[Subsets[Range[n]],Mean[#]==k&]],{n,10},{k,n}]

A359897 Number of strict integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 10, 7, 10, 13, 11, 9, 20, 10, 20, 18, 21, 12, 30, 24, 28, 27, 30, 15, 73, 16, 37, 43, 45, 67, 74, 19, 55, 71, 126, 21, 150, 22, 75, 225, 78, 24, 183, 126, 245, 192, 132, 27, 284, 244, 403, 303, 120, 30, 828

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)
                                                        (5,3,1)

The non-strict version is A240219, complement A359894, ranked by A359889.
The complement is counted by A359898.
The odd-length case is A359899, complement A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A237984 counts partitions containing their mean, complement A327472.
A240850 counts strict partitions containing their mean, complement A240851.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
Cf. A065795, A066571, A067659, A082550, A102627, A135342, A316313, A327473, A327475, A328966, A359909.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Mean[#]==Median[#]&]],{n,0,30}]

A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 42, 72, 124, 218, 390, 706, 1292, 2388, 4436, 8292, 15578, 29376, 55592, 105532, 200858, 383220, 732756, 1403848, 2694404, 5179938, 9973430, 19229826, 37125562, 71762396, 138871260, 269021848, 521666984, 1012520400, 1966957692, 3824240848

Offset: 1

John W. Layman, Dec 05 2001

nonn

Also the number of subsets of {1,2,...,n} with sum of entries divisible by the largest element (compare A000016). See the Palmer Melbane link for a bijection. - Joel B. Lewis, Nov 13 2014

			a(4)=6, since {1}, {2}, {3}, {4}, {1,2,3} and {2,3,4} contain their averages.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(6) = 16 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                     {2,3,4}  {1,2,3}      {6}
                              {1,3,5}      {1,2,3}
                              {2,3,4}      {1,3,5}
                              {3,4,5}      {2,3,4}
                              {1,2,3,4,5}  {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {1,2,3,4,5}
                                           {2,3,4,5,6}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..3333
Palmer Melbane, Combinations under constraint, Art of Problem Solving thread. - _Joel B. Lewis_, Nov 13 2014

Subsets containing n whose mean is an element are A000016.
The version for integer partitions is A237984.
Subsets not containing their mean are A327471.
Cf. A051293, A063776, A066571, A135342, A324736, A325705, A326083, A326836, A327478, A327481.

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] &]; Apply[ Plus, 2^(i/a) * EulerPhi[a]]/i, {i, n}]/2, {n, 34}]
(* second program *)
Table[Length[Select[Subsets[Range[n]],MemberQ[#,Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

a(n) = (1/2)*sum(i=1, n, (1/i)*sumdiv(i, d, if (d%2, 2^(i/d)*eulerphi(d)))); \\ Michel Marcus, Dec 20 2020

from sympy import totient, divisors
def A065795(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1 # Chai Wah Wu, Feb 22 2023

a(n) = (1/2)*Sum_{i=1..n} (f(i) - 1) where f(i) = (1/i) * Sum_{d | i and d is odd} 2^(i/d) * phi(d).
a(n) = (n + A051293(n))/2.
a(n) = 2^n - A327471(n). - Gus Wiseman, Sep 14 2019

Edited and extended by Robert G. Wilson v, Nov 15 2002

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

A327471 Number of subsets of {1..n} not containing their mean.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 48, 102, 214, 440, 900, 1830, 3706, 7486, 15092, 30380, 61100, 122780, 246566, 494912, 992984, 1991620, 3993446, 8005388, 16044460, 32150584, 64414460, 129037790, 258462026, 517641086, 1036616262, 2075721252, 4156096036, 8320912744, 16658202200

Offset: 0

Gus Wiseman, Sep 12 2019

nonn

			The a(1) = 1 through a(5) = 22 subsets:
  {}  {}     {}     {}         {}
      {1,2}  {1,2}  {1,2}      {1,2}
             {1,3}  {1,3}      {1,3}
             {2,3}  {1,4}      {1,4}
                    {2,3}      {1,5}
                    {2,4}      {2,3}
                    {3,4}      {2,4}
                    {1,2,4}    {2,5}
                    {1,3,4}    {3,4}
                    {1,2,3,4}  {3,5}
                               {4,5}
                               {1,2,4}
                               {1,2,5}
                               {1,3,4}
                               {1,4,5}
                               {2,3,5}
                               {2,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}

Subsets containing their mean are A065795.
Subsets containing n but not their mean are A327477.
Partitions not containing their mean are A327472.
Strict partitions not containing their mean are A240851.
Cf. A000016, A007865, A051293, A067538, A082550, A114639, A135342, A324741, A327476.

Table[Length[Select[Subsets[Range[n]],!MemberQ[#,Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327471(n): return (1<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1) # Chai Wah Wu, Feb 22 2023

a(n) = 2^n - A065795(n). - Alois P. Heinz, Sep 13 2019

More terms from Alois P. Heinz, Sep 13 2019

A327474 Number of distinct means of subsets of {1..n}, where {} has mean 0.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 38, 56, 78, 106, 138, 180, 226, 284, 348, 420, 500, 596, 698, 818, 946, 1086, 1236, 1408, 1588, 1788, 2000, 2230, 2472, 2742, 3020, 3328, 3652, 3996, 4356, 4740, 5136, 5568, 6018, 6492, 6982, 7512, 8054, 8638, 9242, 9870, 10520, 11216

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(3) = 6 distinct means are 0, 1, 3/2, 2, 5/2, 3.

The version for only nonempty subsets is A135342.

Cf. A000016, A051293, A065795, A082550, A327475, A327481.

a:= proc(n) option remember; `if`(n<4, [1, 2, 4, 6][n+1],
      2*a(n-1)-a(n-2)+numtheory[phi](n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 22 2023

Table[Length[Union[Mean/@Subsets[Range[n]]]],{n,0,10}]

from itertools import count, islice
from sympy import totient
def A327474_gen(): # generator of terms
    a, b = 4, 6
    yield from (1,2,4,6)
    for n in count(3):
        a, b = b, (b<<1)-a+totient(n)
        yield b
A327474_list = list(islice(A327474_gen(),30)) # Chai Wah Wu, Feb 22 2023

a(n) = A135342(n) + 1.

a(n) = 2*a(n-1) - a(n-2) + phi(n-1) for n>3. - Chai Wah Wu, Feb 22 2023

A327477 Number of subsets of {1..n} containing n whose mean is not an element.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 26, 54, 112, 226, 460, 930, 1876, 3780, 7606, 15288, 30720, 61680, 123786, 248346, 498072, 998636, 2001826, 4011942, 8039072, 16106124, 32263876, 64623330, 129424236, 259179060, 518975176, 1039104990, 2080374784, 4164816708, 8337289456

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(1) = 1 through a(5) = 12 subsets:
  {1,2}  {1,3}  {1,4}      {1,5}
         {2,3}  {2,4}      {2,5}
                {3,4}      {3,5}
                {1,2,4}    {4,5}
                {1,3,4}    {1,2,5}
                {1,2,3,4}  {1,4,5}
                           {2,3,5}
                           {2,4,5}
                           {1,2,3,5}
                           {1,2,4,5}
                           {1,3,4,5}
                           {2,3,4,5}

Subsets whose mean is an element are A065795.
Subsets whose mean is not an element are A327471.
Subsets containing n whose mean is an element are A000016.
Cf. A007865, A051293, A082550, A135342, A240851, A324741, A327472.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!MemberQ[#,Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327477(n): return (1<>(~n&n-1).bit_length(),generator=True))//n if n else 0 # Chai Wah Wu, Feb 21 2023

From Alois P. Heinz, Feb 21 2023: (Start)
a(n) = A327471(n) - A327471(n-1) for n>=1.
a(n) = 2^(n-1) - A000016(n) for n>=1. (End)

a(25)-a(34) from Alois P. Heinz, Feb 21 2023

A327484 Number of integer partitions of 2^n whose mean is a power of 2.

Original entry on oeis.org

1, 2, 4, 11, 66, 1417, 178803, 275379307, 15254411521973, 108800468645440803267, 964567296140908420613296779144, 219614169629364529542990295052656098001967511, 38626966436500261962963100479469496821891576834974275502742922521

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

Number of partitions of 2^n whose number of parts is a power of 2. - Chai Wah Wu, Sep 21 2023

			The a(0) = 1 through a(3) = 11 partitions:
  (1)  (2)   (4)     (8)
       (11)  (22)    (44)
             (31)    (53)
             (1111)  (62)
                     (71)
                     (2222)
                     (3221)
                     (3311)
                     (4211)
                     (5111)
                     (11111111)

Chai Wah Wu, Table of n, a(n) for n = 0..15 (n = 0..13 from Alois P. Heinz)

Row sums of A327483.

Cf. A067538, A068413, A135342, A237984, A327474, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],IntegerQ[Mean[#]]&]],{n,0,5}]

from sympy.utilities.iterables import partitions
def A327484(n): return sum(1 for s,p in partitions(1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327484(n): return sum(A008284_T(1<Chai Wah Wu, Sep 21 2023

a(7) from Chai Wah Wu, Sep 14 2019
a(8)-a(11) from Alois P. Heinz, Sep 21 2023
a(12) from Chai Wah Wu, Sep 21 2023

cp's OEIS Frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

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A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

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A359897 Number of strict integer partitions of n whose parts have the same mean as median.

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A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

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A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

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A327471 Number of subsets of {1..n} not containing their mean.

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A327474 Number of distinct means of subsets of {1..n}, where {} has mean 0.