A063865 -id:A063865 - cp's OEIS frontend

A113037 Number of solutions to +- 1 +- 2 +- .. +- n = 3.

0, 0, 1, 0, 0, 3, 5, 0, 0, 23, 39, 0, 0, 219, 396, 0, 0, 2406, 4435, 0, 0, 28431, 53167, 0, 0, 353500, 667874, 0, 0, 4557831, 8675836, 0, 0, 60382450, 115601178, 0, 0, 816998489, 1571272955, 0, 0, 11242173783, 21701318843, 0, 0, 156841667096

Offset: 0

Floor van Lamoen, Oct 11 2005

nonn

Ray Chandler, Table of n, a(n) for n = 0..3340 (terms < 10^1000)

Cf. A058377, A063865, A063866.

A113037:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-i)+x^i); od; t:=t,coeff(p,x,3); od; t; end;

nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2] + 3;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 14 2014 *)

a(n) is the coefficient of x^3 in product(x^(-k)+x^k, k=1..n).

A293576 Numbers n such that the set of exponents in expression for 2*n as a sum of distinct powers of 2 can be partitioned into two parts with equal sums.

Original entry on oeis.org

0, 7, 13, 15, 22, 25, 27, 30, 39, 42, 45, 47, 49, 51, 54, 59, 60, 62, 75, 76, 82, 85, 87, 90, 93, 95, 97, 99, 102, 107, 108, 110, 117, 119, 120, 122, 125, 127, 141, 143, 147, 148, 153, 155, 158, 162, 165, 167, 170, 173, 175, 179, 180, 185, 187, 188, 190, 193

Offset: 1

Rémy Sigrist, Oct 12 2017

More informally, this sequence encodes finite sets of positive numbers, say { e_1, e_2, ..., e_h }, such that +- e_1 +- e_2 ... +- e_h = 0 has a solution.
The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
No term can have a Hamming weight of 1 or 2.
If x and y belong to this sequence and x AND y = 0 (where AND stands for the bitwise and-operator), then x + y belongs to this sequence.
If k has an odd Hamming weight, then there are only a finite number of terms with the same odd part as k (see A000265 for the odd part of a number).
The number 2^k-1 belongs to this sequence iff A063865(k) > 0.
If k has Hamming weight > 1, then k + 2^(A029931(k)-1) belongs to this sequence.

			2*42 = 2^6 + 2^4 + 2^2 and 6 = 4 + 2, hence 42 appears in the sequence.
2*11 = 2^4 + 2^2 + 2^1 and { 1, 2, 4 } cannot be partitioned into two parts with equals sums, hence 11 does not appear in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000120, A000265, A029931, A133457, A293664.

b:= proc(n, t) option remember; `if`(n=0, is(t=0),
      (i-> b(n-2^i, t-i) or b(n-2^i, t+i))(ilog2(n)))
    end:
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, -1, a(n-1)) while not b(2*k, 0) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 22 2017

is(n) = { my (v=Set(0)); my (b = Vecrev(binary(n))); for (i=1, #b, if (b[i], v = set union(Set(vector(#v, k, v[k]-i)), Set(vector(#v, k, v[k]+i))););); return (set search(v,0)); }

A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 0, 4, 2, 2, 4, 2, 2, 5, 5, 1, 2, 2, 3, 2, 5, 3, 4, 2, 2, 3, 10, 2, 4, 7, 5, 3, 3, 7, 6, 4, 4, 5, 5, 5, 2, 6, 4, 6, 5, 3, 8, 4, 5, 4, 5, 2, 10, 5, 3, 7, 11, 6, 10, 5, 11, 6, 4, 7, 6, 10

Offset: 0

Gus Wiseman, Jul 07 2019

nonn

Also by definition the number of subsets whose sum is one fewer than the product of their complement.

			The initial terms count the following subsets:
   0: {}
   1: {1}
   2: {2}
   3: {1,3}
   4: {2,3}
   7: {4,5}
  10: {1,6,7}
  12: {7,9}
  12: {1,2,4,8}
  14: {2,5,9}
  14: {1,2,4,11}
  15: {1,3,5,7}
  16: {3,4,10}
  16: {1,3,5,8}
  17: {1,10,13}
  18: {2,5,15}
  19: {11,15}
  19: {1,2,6,14}
  20: {1,4,6,8}

Giovanni Resta, Table of n, a(n) for n = 0..2500

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325041, A326172, A326173, A326174, A326175, A326179, A326180, A326441.

Table[Length[Select[Subsets[Range[n]],1+Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
ric[n_, pr_, s_, lst_, t_] := Block[{k}, If[pr == t-s, cnt++]; Do[ If[pr k <= t, ric[n, pr k, s + k, k, t], Break[]], {k, lst+1, n}]]; a[n_] := (cnt = 0; ric[n, 1, 0, 0, n (n + 1)/2 + 1]; cnt); a /@ Range[0, 85] (* Giovanni Resta, Sep 13 2019 *)

More terms from Alois P. Heinz, Jul 12 2019

A342602 Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 6, 14, 29, 63, 129, 300, 756, 1677, 4134, 9525, 22841, 57175, 141819, 354992, 882420, 2218078, 5588989, 14173217, 35918542

Offset: 1

Scott R. Shannon, Mar 27 2021

Normal operator precedence is followed, so multiplication is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 3.

The author thanks Ursula Ponting for useful discussions.

			a(3) = 1 as 1 + 2 - 3 = 0 is the only solution.
a(4) = 1 as 1 - 2 - 3 + 4 = 0 is the only solution.
a(5) = 1 as 1 * 2 - 3 - 4 + 5 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(6) = 4. The solutions, all of which use multiplication, are
         1 + 2 * 3 + 4 - 5 - 6 = 0,
         1 - 2 + 3 * 4 - 5 - 6 = 0,
         1 - 2 * 3 + 4 - 5 + 6 = 0,
         1 * 2 + 3 - 4 + 5 - 6 = 0.
a(10) = 63. An example solution is
         1 - 2 * 3 * 4 - 5 - 6 - 7 * 8 + 9 * 10 = 0.
a(20) = 354992. An example solution is
         1 * 2 * 3 * 4 * 5 * 6 * 7 + 8 * 9 + 10 * 11 - 12 * 13 + 14 * 15
             - 16 * 17 * 18 - 19 * 20 = 0
  which includes thirteen multiplications.

Cf. A342804 (using +-*/), A342995 (using +-/), A058377 (using +-), A063865, A000217, A025591, A161943.

Table[Length@Select[Tuples[{"+","-","*"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from itertools import product
def a(n):
  nn = [str(i) for i in range(1, n+1)]
  return sum(eval("".join([*sum(zip(nn, ops+("",)), ())])) == 0 for ops in product("+-*", repeat=n-1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Apr 02 2021

A342995 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 4, 8, 0, 3, 37, 80, 6, 17, 461, 868, 190, 364, 5570, 11342, 3993, 7307, 78644

Offset: 1

Scott R. Shannon, Apr 01 2021

Normal operator precedence is followed, so division is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 10.

			a(3) = 1 as 1 + 2 - 3 = 0 is the only solution.
a(4) = 1 as 1 - 2 - 3 + 4 = 0 is the only solution.
a(5) = 0, as in A058377.
a(6) = 1 as 1 - 2 / 3 / 4 - 5 / 6 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(8) = 8. Seven of the solutions involve just addition and subtraction, matching those in A058377, but one additional solution exists using division:
        1 / 2 / 3 / 4 + 5 / 6 - 7 / 8 = 0.
a(10) = 3. All three solutions require division:
        1 + 2 / 3 / 4 + 5 / 6 + 7 - 8 + 9 - 10 = 0,
        1 - 2 / 3 / 4 - 5 / 6 + 7 - 8 - 9 + 10 = 0,
        1 - 2 / 3 / 4 - 5 / 6 - 7 + 8 + 9 - 10 = 0.
a(15) = 461. Of these, 361 use only addition and subtraction, the other 100 also require division. One example of the latter is
        1 / 2 / 3 / 4 - 5 - 6 - 7 / 8 + 9 / 10 + 11 + 12 - 13 + 14 / 15 = 0.
a(20) = 11342. An example solution is
        1 / 2 / 3 - 4 / 5 / 6 + 7 / 8 / 9 + 10 + 11 / 12 - 13 + 14 / 15 / 16
             + 17 / 18 + 19 / 20 = 0
  which sums seven fractions that include eleven divisions.

Cf. A342804 (using +-*/), A342602 (using +-*), A058377 (using +-), A063865, A000217, A025591, A161943.

Table[Length@Select[Tuples[{"+","-","/"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from itertools import product
from fractions import Fraction
def a(n):
  nn = ["Fraction("+str(i)+", 1)" for i in range(1, n+1)]
  return sum(eval("".join([*sum(zip(nn, ops+("",)), ())])) == 0 for ops in product("+-/", repeat=n-1))
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Apr 02 2021

A369873 a(n) is the constant term in the expansion of Product_{d|n} (x^d + 1/x^d).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 34, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 28, 0, 0, 0, 2, 0, 26, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 4, 0, 2, 0

Offset: 1

Ilya Gutkovskiy, Feb 03 2024

nonn

a(n) is the number of solutions to 0 = Sum_{d|n} c_i * d with c_i in {-1,1}, i=1..tau(n), tau = A000005.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A033630, A063865, A083206, A083207, A379504.

Table[Coefficient[Product[(x^d + 1/x^d), {d, Divisors[n]}], x, 0], {n, 1, 90}]

A369873(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); polcoeff(p, 0)); }; \\ (cf. also code in A083206 and A379504) - Antti Karttunen, Jan 20 2025

from collections import Counter
from sympy import divisors
def A369873(n):
    c = {0:1}
    for d in divisors(n,generator=True):
        b = Counter()
        for j in c:
            a = c[j]
            b[j+d] += a
            b[j-d] += a
        c = b
    return c[0] # Chai Wah Wu, Feb 05 2024

From Joerg Arndt, Feb 04 2024: (Start)
a(n) != 0 (only) for n in A083207.
a(n) = 2 * A083206(n). (End)

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A379451 Number of ordered ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

Original entry on oeis.org

2, 10, 162, 6278, 430906, 46032666, 7029940154, 1453778429782, 390651831405906, 132345369222827306, 55150093300481888770, 27727437337790844360198, 16545310955942988999292586, 11561068480810074519638819626, 9349537740123803513263001013354, 8664632430514446774520557369434870

Offset: 0

Ilya Gutkovskiy, Dec 23 2024

nonn

			a(1) = 10 ways: {}, {0}, {-1, 1} (2 permutations), {-1, 0, 1} (6 permutations).

Michael S. Branicky, Table of n, a(n) for n = 0..100

Cf. A000980, A058377, A063865.

from math import factorial
from functools import cache
@cache
def b(i, s, c):
    if i == 0: return factorial(c) if s == 0 else 0
    return b(i-1, s, c) + b(i-1, s+(i>>1)*(-1)**(i&1), c+1)
def a(n): return b(2*n+1, 0, 0)
print([a(n) for n in range(16)]) # Michael S. Branicky, Dec 23 2024

a(9) and beyond from Michael S. Branicky, Dec 23 2024

A174600 T(n,k) = 1 if the sum of +-k..+-n with arbitrary signs never equals zero, = 0 otherwise (lower triangle).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1

Offset: 1

R. H. Hardin Mar 23 2010, with formula and reference from Franklin T. Adams-Watters and Olivier Gérard, on the Sequence Fans Mailing list

			Triangle begins
1;
1, 1;
0, 1, 1;
0, 1, 1, 1;
1, 0, 1, 1, 1;
1, 0, 0, 1, 1, 1;
0, 1, 1, 0, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1, 1;
1, 0, 0, 1, 1, 0, 1, 1, 1;
1, 0, 0, 1, 1, 1, 0, 1, 1, 1; ...

R. H. Hardin, Table of n, a(n) for n=1..4950

Related to A063865.

{ for(n=1; n<10; n++)
        for(k=1; k<=n; k++)
            print ++i, T(n,k);
}
function T(n,k) {
    if ( int((n+1)/2)%2 != int(k/2)%2 ) return 1;
    else if ( (n-k)%2 == 0 ) {
        if ( k > ((n-k)/2)^2 ) return 1;
        else return 0;
    }
    else return 0;
}

t[n_, k_] := If[Mod[Floor[(n+1)/2], 2] != Mod[Floor[k/2], 2], 1, If[Mod[n-k, 2] == 0, If[k > ((n-k)/2)^2, 1, 0], 0]]; Flatten[Table[t[n, k], {n, 1, 15}, {k, 1, n}]][[;; 108]] (* Jean-François Alcover, Jul 11 2011, after awk program *)

T(n,k)=
{
    if ( ((n+1)\2)%2 != (k\2)%2,
        return(1);
    , /* else */
        if ( (n-k)%2 == 0,
            if ( k > ((n-k)/2)^2, return(1), return(0) );
        , /* else */
            return(0);
        );
    );
}
{ for(n=1, 10, /* show triangle */
    for(k=1,n,
        print1(T(n,k),", ");
    );
    print();
 ); }

cp's OEIS Frontend

A113037 Number of solutions to +- 1 +- 2 +- .. +- n = 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A293576 Numbers n such that the set of exponents in expression for 2*n as a sum of distinct powers of 2 can be partitioned into two parts with equal sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A342602 Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A342995 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A369873 a(n) is the constant term in the expansion of Product_{d|n} (x^d + 1/x^d).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A379451 Number of ordered ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A174600 T(n,k) = 1 if the sum of +-k..+-n with arbitrary signs never equals zero, = 0 otherwise (lower triangle).

Views

Author

Keywords