A222955 -id:A222955 - cp's OEIS frontend

A000980 Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

2, 4, 8, 20, 52, 152, 472, 1520, 5044, 17112, 59008, 206260, 729096, 2601640, 9358944, 33904324, 123580884, 452902072, 1667837680, 6168510256, 22903260088, 85338450344, 318995297200, 1195901750512, 4495448217544, 16940411201280, 63983233268592

Offset: 0

N. J. A. Sloane

The 4-term sequence 2,4,8,20 is the answer to the "Solitaire Army" problem, or checker-jumping puzzle. It is too short to have its own entry. See Conway et a;., Winning Ways, Vol. 2, pp. 715-717. - N. J. A. Sloane, Mar 01 2018

Number of subsets of {-n..n} with sum 0. Also the number of subsets of {0..2n} that are empty or have mean n. For median instead of mean we have twice A024718. - Gus Wiseman, Apr 23 2023

			From _Gus Wiseman_, Apr 23 2023: (Start)
The a(0) = 2 through a(2) = 8 subsets of {-n..n} with sum 0 are:
  {}   {}        {}
  {0}  {0}       {0}
       {-1,1}    {-1,1}
       {-1,0,1}  {-2,2}
                 {-1,0,1}
                 {-2,0,2}
                 {-2,-1,1,2}
                 {-2,-1,0,1,2}
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 715-717.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 0..1668 (terms < 10^1000; terms 0..200 from T. D. Noe, terms 201..400 from Alois P. Heinz)
Eunice Y. S. Chan and R. M. Corless, Narayana, Mandelbrot, and A New Kind of Companion Matrix, arXiv preprint arXiv:1606.09132 [math.CO], 2016.
R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
J. H. van Lint, Representations of 0 as Sum_{k = -N..N} epsilon_k*k, Proc. Amer. Math. Soc., 18 (1967), 182-184.

A047653(n) = a(n)/2.
Bisection of A084239. Cf. A063865, A141000.
A007318 counts subsets by length, A327481 by integer mean.
A327475 counts subsets with integer mean, A000975 integer median.
Cf. A024718, A047997, A070925, A079309, A133406, A212352, A359893, A362046.

a000980 n = length $ filter ((== 0) . sum) $ subsequences [-n..n]

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> 2*b(0, n):
seq(a(n), n=0..40); # Alois P. Heinz, Mar 10 2014

a[n_] := SeriesCoefficient[ Product[1+x^k, {k, -n, n}], {x, 0, 0}]; a[0] = 2; Table[a[n], {n, 0, 24}](* Jean-François Alcover, Nov 28 2011 *)
nmax = 26; d = {2}; a1 = {};
Do[
  i = Ceiling[Length[d]/2];
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n] +
    2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 15 2014 *)
Table[Length[Select[Subsets[Range[-n,n]],Total[#]==0&]],{n,0,5}] (* Gus Wiseman, Apr 23 2023 *)

PARI
```
a(n)=polcoeff(prod(k=-n,n,1+x^k),0)
```

Constant term of Product_{k=-n..n} (1+x^k).
a(n) = Sum_i A067059(2n+1-i, i) = 2+2*Sum_j A047997(n, j); i.e., sum of alternate antidiagonals of A067059 and two more than twice row sums of A047997. - Henry Bottomley, Aug 11 2002
a(n) = A004171(n) - 2*A181765(n).
Coefficient of x^(n*(n+1)/2) in 2*Product_{k=1..n} (1+x^k)^2. - Sean A. Irvine, Oct 03 2011
From Gus Wiseman, Apr 23 2023: (Start)
a(n) = 2*A047653(n).
a(n) = A070925(2n+1) + 1.
a(n) = 2*A133406(2n+1).
a(n) = 2*(A212352(n) + 1).
a(n) = A222955(2n+1).
a(n) = 2*(A362046(2n) + 1).
(End)

More terms from Michael Somos, Jun 10 2000

A359402 Numbers whose binary expansion and reversed binary expansion have the same sum of positions of 1's, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 70, 73, 78, 85, 93, 99, 107, 119, 127, 129, 150, 153, 165, 189, 195, 219, 231, 255, 257, 266, 273, 282, 294, 297, 310, 313, 325, 334, 341, 350, 355, 365, 371, 381, 387, 397, 403, 413, 427, 443, 455, 471

Offset: 1

Gus Wiseman, Jan 05 2023

nonn

Also numbers whose binary expansion and reversed binary expansion have the same sum of partial sums.
Also numbers whose average position of a 1 in their binary expansion is (c+1)/2, where c is the number of digits.
Conjecture: Also numbers whose binary expansion has as least squares fit a line of zero slope, counted by A222955.

			The binary expansion of 70 is (1,0,0,0,1,1,0), with positions of 1's {1,5,6}, while the reverse positions are {2,3,7}. Both sum to 12, so 70 is in the sequence.

Binary words of this type appear to be counted by A222955.
For greater instead of equal sums we have A359401.
These are the indices of 0's in A359495.
A030190 gives binary expansion, reverse A030308.
A048793 lists partial sums of reversed standard compositions, sums A029931.
A070939 counts binary digits, 1's A000120.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
Cf. A051293, A053632, A231204, A291166, A304818, A318283, A326672, A326673, A358134, A359042.

Select[Range[0,100],#==0||Mean[Join@@Position[IntegerDigits[#,2],1]]==(IntegerLength[#,2]+1)/2&]

from functools import reduce
from itertools import count, islice
def A359402_gen(startvalue=0): # generator of terms
    return filter(lambda n:(r:=reduce(lambda c, d:(c[0]+d[0]*(e:=int(d[1])),c[1]+e),enumerate(bin(n)[2:],start=1),(0,0)))[0]<<1==(n.bit_length()+1)*r[1],count(max(startvalue,0)))
A359402_list = list(islice(A359402_gen(),30)) # Chai Wah Wu, Jan 08 2023

A230877(a(n)) = A029931(a(n)).

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

0, 0, -1, 0, -2, 0, -2, 0, -3, 0, -2, 1, -4, -1, -3, 0, -4, 0, -2, 2, -4, 0, -2, 2, -6, -2, -4, 0, -6, -2, -4, 0, -5, 0, -2, 3, -4, 1, -1, 4, -6, -1, -3, 2, -5, 0, -2, 3, -8, -3, -5, 0, -7, -2, -4, 1, -9, -4, -6, -1, -8, -3, -5, 0, -6, 0, -2, 4, -4, 2, 0, 6

Offset: 0

Gus Wiseman, Jan 05 2023

Also the sum of partial sums of reversed binary expansion minus sum of partial sums of binary expansion.

			The binary expansion of 158 is (1,0,0,1,1,1,1,0), with positions of 1's {1,4,5,6,7} with sum 23, reversed {2,3,4,5,8} with sum 22, so a(158) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Indices of positive terms are A359401.
Indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
Cf. A000120, A048793, A053632, A222955, A231204, A291166, A326669, A326672, A326673, A359042, A359043.

a:= n-> (l-> add(i*(l[-i]-l[i]), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..127);  # Alois P. Heinz, Jan 09 2023

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Table[sap[IntegerDigits[n,2]],{n,0,100}]

def A359495(n):
    k = n.bit_length()-1
    return sum((i<<1)-k for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

a(n) = A029931(n) - A230877(n).

If n = Sum_{i=1..k} q_i * 2^(i-1), then a(n) = Sum_{i=1..k} q_i * (2i-k-1).

A070925 Number of subsets of A = {1,2,...,n} that have the same center of gravity as A, i.e., (n+1)/2.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 19, 17, 51, 47, 151, 137, 471, 427, 1519, 1391, 5043, 4651, 17111, 15883, 59007, 55123, 206259, 193723, 729095, 688007, 2601639, 2465133, 9358943, 8899699, 33904323, 32342235, 123580883, 118215779, 452902071, 434314137, 1667837679, 1602935103

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 20 2002

nonn

From Gus Wiseman, Apr 15 2023: (Start)
Also the number of nonempty subsets of {0..n} with mean n/2. The a(0) = 1 through a(5) = 7 subsets are:
  {0}  {0,1}  {1}      {0,3}      {2}          {0,5}
              {0,2}    {1,2}      {0,4}        {1,4}
              {0,1,2}  {0,1,2,3}  {1,3}        {2,3}
                                  {0,2,4}      {0,1,4,5}
                                  {1,2,3}      {0,2,3,5}
                                  {0,1,3,4}    {1,2,3,4}
                                  {0,1,2,3,4}  {0,1,2,3,4,5}
(End)

			Of the 32 (2^5) sets which can be constructed from the set A = {1,2,3,4,5} only the sets {3}, {2, 3, 4}, {2, 4}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 5} give an average of 3.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..40

The odd bisection is A000980(n) - 1 = 2*A047653(n) - 1.
For median instead of mean we have A100066, bisection A006134.
Including the empty set gives A222955.
The one-based version is A362046, even bisection A047653(n) - 1.
A007318 counts subsets by length.
A067538 counts partitions with integer mean, strict A102627.
A231147 counts subsets by median.
A327481 counts subsets by integer mean.
Cf. A024718, A057552, A079309, A133406, A326512, A326513, A326537, A327475, A349156, A359893.

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{s = Subsets[n], c = 0, k = 2}, While[k < 2^n + 1, If[ (Plus @@ s[[k]]) / Length[s[[k]]] == (n + 1)/2, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 20}]
(* second program *)
Table[Length[Select[Subsets[Range[0,n]],Mean[#]==n/2&]],{n,0,10}] (* Gus Wiseman, Apr 15 2023 *)

From Gus Wiseman, Apr 18 2023: (Start)
a(2n+1) = A000980(n) - 1.
a(n) = A222955(n) - 1.
a(n) = 2*A362046(n) + 1.
(End)

Edited by Robert G. Wilson v and John W. Layman, May 25 2002

a(34)-a(38) from Fausto A. C. Cariboni, Oct 08 2020

A222970 Number of 1 X (n+1) 0..1 arrays with every row least squares fitting to a positive-slope straight line and every column least squares fitting to a zero- or positive-slope straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

1, 2, 6, 12, 28, 54, 119, 230, 488, 948, 1979, 3860, 7978, 15624, 32072, 63014, 128746, 253588, 516346, 1019072, 2069590, 4091174, 8291746, 16412668, 33210428, 65808044, 132985161, 263755984, 532421062, 1056789662, 2131312530, 4233176854

Offset: 1

R. H. Hardin, Mar 10 2013

nonn

From Gus Wiseman, Jun 16 2023: (Start)
Also appears to be the number of integer compositions of n + 2 with weighted sum greater than reverse-weighted sum, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i, and the reverse is Sum_{i=1..k} i * y_{k-i+1}. The a(1) = 1 through a(4) = 12 compositions are:
  (21)  (31)   (32)    (42)
        (211)  (41)    (51)
               (221)   (231)
               (311)   (312)
               (1211)  (321)
               (2111)  (411)
                       (1311)
                       (2121)
                       (2211)
                       (3111)
                       (12111)
                       (21111)
The version for partitions is A144300, strict A111133.
(End)

			Some solutions for n=3:
  0 1 0 1    0 1 1 1    0 0 1 0    0 0 1 1    0 0 0 1

R. H. Hardin, Table of n, a(n) for n = 1..210

For >= instead of > we have A222855.
The case of equality is A222955.
Row 1 of A222969.
A053632 counts compositions by weighted sum (or  reverse-weighted sum).
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, reverse A318283.
Cf. A000005, A000041, A138364, A320387, A360672, A360675, A363626.

A231429 Number of partitions of 2n into distinct parts < n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 4, 8, 14, 22, 35, 53, 78, 113, 160, 222, 306, 416, 558, 743, 980, 1281, 1665, 2149, 2755, 3514, 4458, 5626, 7070, 8846, 11020, 13680, 16920, 20852, 25618, 31375, 38309, 46649, 56651, 68616, 82908, 99940, 120192, 144238, 172730, 206425

Offset: 0

Reinhard Zumkeller, Nov 14 2013

nonn

From Gus Wiseman, Jun 17 2023: (Start)
Also the number of integer compositions of n with weighted sum 3*n, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i. The a(0) = 1 through a(9) = 14 compositions are:
  ()  .  .  .  .  (11111)  (3111)   (3211)   (3311)    (3411)
                           (11211)  (11311)  (4121)    (4221)
                                    (12121)  (11411)   (5112)
                                    (21112)  (12221)   (11511)
                                             (13112)   (12321)
                                             (21131)   (13131)
                                             (21212)   (13212)
                                             (111122)  (21231)
                                                       (21312)
                                                       (22122)
                                                       (31113)
                                                       (111141)
                                                       (111222)
                                                       (112113)
For partitions we have A363527, ranks A363531. For reversed partitions we have A363526, ranks A363530.
(End)

			a(5) = #{4+3+2+1} = 1;
a(6) = #{5+4+3, 5+4+2+1} = 2;
a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
a(8) = #{7+6+3, 7+6+2+1, 7+6+3, 7+5+3+1, 7+4+3+2, 6+5+4+1, 6+5+3+2, 6+4+3+2+1} = 8;
a(9) = #{8+7+3, 8+7+2+1, 8+6+4, 8+6+3+1, 8+5+4+1, 8+5+3+2, 8+4+3+2+1, 7+6+5, 7+6+4+1, 7+6+3+2, 7+5+4+2, 7+5+3+2+1, 6+5+4+3, 6+5+4+2+1} = 14.

Cf. A209815, A079122.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A008284, A029931, A067538, A222855, A222955, A222970, A359042, A360672, A360675, A362559.

a231429 n = p [1..n-1] (2*n) where
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]],{n,0,15}] (* Gus Wiseman, Jun 17 2023 *)

A245134 T(n,k)=Number of length n 0..k arrays least squares fitting to a zero slope straight line, with a single point taken as having zero slope.

Original entry on oeis.org

2, 3, 2, 4, 3, 4, 5, 4, 9, 4, 6, 5, 16, 9, 8, 7, 6, 25, 22, 39, 8, 8, 7, 36, 41, 112, 43, 20, 9, 8, 49, 66, 275, 172, 195, 18, 10, 9, 64, 107, 552, 505, 1064, 243, 52, 11, 10, 81, 158, 1029, 1248, 4005, 1742, 1209, 48, 12, 11, 100, 219, 1728, 2687, 11856, 8193, 11664, 1539

Offset: 1

R. H. Hardin, Jul 12 2014

Table starts
..2....3.....4......5......6.......7.......8........9.......10........11
..2....3.....4......5......6.......7.......8........9.......10........11
..4....9....16.....25.....36......49......64.......81......100.......121
..4....9....22.....41.....66.....107.....158......219......304.......403
..8...39...112....275....552....1029....1728.....2781.....4200......6171
..8...43...172....505...1248....2687....5220.....9385....15868.....25539
.20..195..1064...4005..11856...29813...66256...134091...252060....446193
.18..243..1742...8193..29182...85529..217336...494943..1033716...2012883
.52.1209.11664..68855.294024.1006089.2920784..7483887.17365380..37197259
.48.1539.19976.147117.754712.3011889.9995864.28810117.74285448.175024363

			Some solutions for n=7 k=4
..3....1....2....0....4....0....2....3....0....4....3....1....0....2....4....1
..1....4....0....0....0....1....1....0....3....0....3....3....2....1....0....3
..2....4....2....4....4....2....3....0....4....2....0....1....3....0....4....0
..3....2....4....0....1....2....4....4....3....3....2....4....1....2....4....1
..2....3....3....1....2....1....3....0....4....0....4....4....0....1....0....0
..1....3....1....0....1....0....1....3....0....1....1....3....2....2....2....0
..3....2....1....1....4....1....2....1....2....4....3....0....1....1....4....3

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A222955, terms 1,3,5... are A000980
Column 2 is A223743
Column 3 is A223819
Row 1 is A000027(n+1)
Row 2 is A000027(n+1)
Row 3 is A000290(n+1)
Row 4 is A200155

Empirical for row n:
n=1: a(n) = 2*a(n-1) -a(n-2)
n=2: a(n) = 2*a(n-1) -a(n-2)
n=3: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3)
n=4: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8)
n=5: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)
n=6: [order 18]
n=7: [order 16]

A245851 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends, and least squares fitting to a straight line with slope zero, with a single point taken as having zero slope.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 4, 1, 1, 2, 2, 5, 4, 1, 1, 2, 2, 5, 5, 8, 1, 1, 2, 2, 5, 5, 14, 8, 1, 1, 2, 2, 5, 5, 15, 16, 20, 1, 1, 2, 2, 5, 5, 15, 17, 65, 18, 1, 1, 2, 2, 5, 5, 15, 17, 77, 77, 52, 1, 1, 2, 2, 5, 5, 15, 17, 78, 101, 356, 48, 1, 1, 2, 2, 5, 5, 15, 17, 78, 102, 551, 448, 152

Offset: 1

R. H. Hardin, Aug 04 2014

Table starts
....1......1.......1.......1.......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.......2.......2.......2.......2.......2.......2.......2.......2
....4......5.......5.......5.......5.......5.......5.......5.......5.......5
....4......5.......5.......5.......5.......5.......5.......5.......5.......5
....8.....14......15......15......15......15......15......15......15......15
....8.....16......17......17......17......17......17......17......17......17
...20.....65......77......78......78......78......78......78......78......78
...18.....77.....101.....102.....102.....102.....102.....102.....102.....102
...52....356.....551.....568.....569.....569.....569.....569.....569.....569
...48....448.....861.....918.....919.....919.....919.....919.....919.....919
..152...2279....5433....6115....6142....6143....6143....6143....6143....6143
..138...2959....9055...11063...11170...11171...11171...11171...11171...11171
..472..15572...61725...84095...86043...86081...86082...86082...86082...86082
..428..20762..107467..167139..174957..175169..175170..175170..175170..175170
.1520.111641..758905.1377562.1495234.1500038.1500088.1500089.1500089.1500089
.1392.151205.1355573.2919818.3338570.3364000.3364342.3364343.3364343.3364343

			Some solutions for n=12 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....1....1....1....1....1....1....0....1....1....0....1....1....1....1....0
..1....1....0....0....2....1....2....1....0....2....1....2....0....2....0....1
..2....0....0....0....1....2....3....1....2....1....1....2....0....2....2....1
..0....2....1....1....1....2....1....2....3....3....2....3....0....1....2....1
..1....3....2....2....3....3....3....0....2....1....3....2....1....1....0....0
..0....2....0....1....2....0....0....1....3....2....2....3....2....2....0....0
..2....0....0....2....3....3....2....1....2....1....0....1....1....3....2....1
..1....0....1....0....0....2....3....1....1....2....2....3....1....2....2....1
..1....2....0....1....2....1....2....0....1....2....0....2....0....1....0....1
..0....1....1....0....1....1....1....1....1....1....1....1....0....1....1....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..1121

Column 1 is A222955(n-2)

A359401 Nonnegative integers whose sum of positions of 1's in their binary expansion is greater than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

11, 19, 23, 35, 37, 39, 43, 47, 55, 67, 69, 71, 75, 77, 79, 83, 87, 91, 95, 103, 111, 131, 133, 134, 135, 137, 139, 141, 142, 143, 147, 149, 151, 155, 157, 158, 159, 163, 167, 171, 173, 175, 179, 183, 187, 191, 199, 203, 207, 215, 223, 239, 259, 261, 262, 263

Offset: 1

Gus Wiseman, Jan 05 2023

First differs from A161601 in having 134, with binary expansion (1,0,0,0,0,1,1,0), positions of 1's 1 + 6 + 7 = 14, reversed 2 + 3 + 8 = 13.

Indices of positive terms in A359495; indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
Cf. A000120, A048793, A051293, A053632, A222955, A231204, A291166, A304818, A326672, A326673, A359043.

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Select[Range[0,100],sap[IntegerDigits[#,2]]>0&]

A230877(a(n)) > A029931(a(n)).

A222959 T(n,k)=Number of nXk 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 4, 4, 4, 4, 8, 4, 16, 4, 8, 8, 8, 16, 16, 8, 8, 20, 8, 64, 16, 64, 8, 20, 18, 20, 64, 64, 64, 64, 20, 18, 52, 18, 400, 64, 512, 64, 400, 18, 52, 48, 52, 324, 400, 512, 512, 400, 324, 52, 48, 152, 48, 2704, 324, 8000, 512, 8000, 324, 2704, 48, 152, 138, 152

Offset: 1

R. H. Hardin Mar 10 2013

Table starts
...2...2.....4.....4.......8.......8.........20........18...........52
...2...2.....4.....4.......8.......8.........20........18...........52
...4...4....16....16......64......64........400.......324.........2704
...4...4....16....16......64......64........400.......324.........2704
...8...8....64....64.....512.....512.......8000......5832.......140608
...8...8....64....64.....512.....512.......8000......5832.......140608
..20..20...400...400....8000....8000.....323600....215892.....14843920
..18..18...324...324....5832....5832.....215892....144930......9546228
..52..52..2704..2704..140608..140608...14843920...9546228...1808243216
..48..48..2304..2304..110592..110592....9734400...6503472...1052045568
.152.152.23104.23104.3511808.3511808.1138729280.685380312.407454282560
.138.138.19044.19044.2628072.2628072..631004724.419107674

			Some solutions for n=4 k=4
..0..0..0..0....0..0..0..0....1..1..1..1....1..1..1..1....0..1..1..0
..0..1..1..0....1..1..1..1....0..1..1..0....1..1..1..1....1..1..1..1
..0..1..1..0....1..1..1..1....0..1..1..0....1..1..1..1....1..1..1..1
..0..0..0..0....0..0..0..0....1..1..1..1....1..1..1..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 2 is A222955, column 4 is A222956, column 6 is A222957

cp's OEIS Frontend

A000980 Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A359402 Numbers whose binary expansion and reversed binary expansion have the same sum of positions of 1's, where positions in a sequence are read starting with 1 from the left.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A070925 Number of subsets of A = {1,2,...,n} that have the same center of gravity as A, i.e., (n+1)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A222970 Number of 1 X (n+1) 0..1 arrays with every row least squares fitting to a positive-slope straight line and every column least squares fitting to a zero- or positive-slope straight line, with a single point array taken as having zero slope.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A231429 Number of partitions of 2n into distinct parts < n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Mathematica

A245134 T(n,k)=Number of length n 0..k arrays least squares fitting to a zero slope straight line, with a single point taken as having zero slope.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula