A158092 -id:A158092 - cp's OEIS frontend

A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

1, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 15, 0, 0, 127, 184, 0, 0, 819, 1382, 0, 0, 9441, 18176, 0, 0, 96562, 172371, 0, 0, 1192142, 2252342, 0, 0, 13869696, 25741462, 0, 0, 177056022, 334176492, 0, 0, 2207693292, 4182801839, 0, 0, 28966597122, 55125154468

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A063890, A158092, A348165.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^2), i-1) +b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^2, n)):
seq(a(n), n=0..49);  # Alois P. Heinz, Jan 22 2024

a(n) = [x^(n^2)] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).

A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n(3n-1)/2 = 0.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 4, 4, 0, 30, 46, 78, 0, 210, 366, 644, 0, 2032, 3696, 6694, 0, 21936, 39886, 73098, 0, 246172, 454074, 841714, 0, 2899542, 5401222, 10073398, 0, 35282910, 66213604, 124427582, 0, 441326270, 832775792, 1573861942, 0, 5642205488

Offset: 0

Seiichi Manyama, Sep 17 2017

nonn

			For n=6 the 2 solutions are +1+5-12+22+35-51 = 0 and -1-5+12-22-35+51 = 0.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A000326, A002411, A158092, A158380, A292475.

{a(n) = polcoeff(prod(k=1, n, x^(k*(3*k-1)/2)+1/x^(k*(3*k-1)/2)), 0)}

Constant term in the expansion of Product_{k=1..n} (x^(k*(3*k-1)/2)+1/x^(k*(3*k-1)/2)).

a(4*k+1) = 0 for k >= 0.

A307877 Number of ways of partitioning the set of the first n positive squares into two subsets whose sums differ at most by 1.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 2, 1, 1, 5, 2, 1, 5, 13, 43, 43, 57, 193, 274, 239, 430, 1552, 3245, 2904, 5419, 18628, 31048, 27813, 50213, 188536, 372710, 348082, 649300, 2376996, 4197425, 3913496, 7287183, 27465147, 53072709, 50030553, 93696497, 351329160, 650125358

Offset: 0

Alois P. Heinz, Jun 04 2019

nonn

			a(6) = 2: 1,9,36/4,16,25; 1,4,16,25/9,36.
a(7) = 1: 1,4,16,49/9,25,36.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition problem

Cf. A000290, A069918, A083527, A158092, A306443, A308682.

s:= proc(n) s(n):= `if`(n=0, 1, n^2+s(n-1)) end:
b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
      `if`(n>s(i), 0, (p-> b(n+p, i-1)+b(abs(n-p), i-1))(i^2)))
    end:
a:= n-> ceil(b(0, n)/2):
seq(a(n), n=0..45);

s[n_] := s[n] = If[n == 0, 1, n^2 + s[n - 1]];
b[n_, i_] := b[n, i] = If[i == 0, If[n <= 1, 1, 0], If[n > s[i], 0, Function[p, b[n + p, i - 1] + b[Abs[n - p], i - 1]][i^2]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = A083527(n) if n == 0 or 3 (mod 4).

A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

Original entry on oeis.org

1, 2, 4, 8, 16, 40, 88, 222, 570, 1564, 4516, 13874, 41866, 137432, 442964, 1492610, 4998674, 17204844, 59175316, 207299554, 727137516, 2582078416, 9179001124, 32943918428, 118453240846, 428937325964, 1556421977612, 5676923326262, 20754245720206

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000980, A047653, A158092, A350249, A350881.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1/x^(n^2))^2*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 28 2022

Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Conjecture: a(n) ~ sqrt(5) * 4^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Feb 05 2022

A350403 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 0 or 1.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 2, 2, 2, 5, 2, 2, 10, 13, 43, 86, 114, 193, 274, 478, 860, 1552, 3245, 5808, 10838, 18628, 31048, 55626, 100426, 188536, 372710, 696164, 1298600, 2376996, 4197425, 7826992, 14574366, 27465147, 53072709, 100061106, 187392994, 351329160

Offset: 0

Ilya Gutkovskiy, Dec 29 2021

nonn

			a(8) = 2: +1^2 - 2^2 - 3^2 + 4^2 - 5^2 + 6^2 + 7^2 - 8^2 =
          -1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2 - 7^2 + 8^2 = 0.

Cf. A000290, A025591, A083527, A158092.

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(2*i+1)/6, 0,
      `if`(i=0, 1, b(n+i^2, i-1)+b(abs(n-i^2), i-1)))
    end:
a:=n-> b(0, n)+b(1, n):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 16 2022

b[n_, i_] := b[n, i] = If[n > i*(i + 1)*(2*i + 1)/6, 0,
     If[i == 0, 1, b[n + i^2, i - 1] + b[Abs[n - i^2], i - 1]]];
a[n_] := b[0, n] + b[1, n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

from itertools import product
def a(n):
    if n == 0: return 1
    nn = ["0"] + [str(i)+"**2" for i in range(1, n+1)]
    return sum(eval("".join([*sum(zip(nn, ops+("", )), ())])) in {0, 1} for ops in product("+-", repeat=n))
print([a(n) for n in range(18)]) # Michael S. Branicky, Jan 16 2022

from functools import cache
@cache
def b(n, i):
    if n > i*(i+1)*(2*i+1)//6: return 0
    if i == 0: return 1
    return b(n+i**2, i-1) + b(abs(n-i**2), i-1)
def a(n): return b(0, n) + b(1, n)
print([a(n) for n in range(43)]) # Michael S. Branicky, Jan 16 2022 after Alois P. Heinz

A369730 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 0, 0, 5, 2, 0, 0, 13, 43, 0, 0, 193, 274, 0, 0, 1552, 3245, 0, 0, 18628, 31048, 0, 0, 188536, 372710, 0, 0, 2376996, 4197425, 0, 0, 27465147, 53072709, 0, 0, 351329160, 650125358, 0, 0, 4398613111, 8429649875, 0, 0, 57629346805, 108986428106

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A000290, A063866, A158092, A348165, A350403, A369731, A369732.

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(2*i+1)/6, 0,
      `if`(i=0, 1, b(n+i^2, i-1)+b(abs(n-i^2), i-1)))
    end:
a:=n-> b(1, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2024

Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 48}]

a(n) = [x^1] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).

A292510 a(n) = smallest k >= 1 such that {1, p(n,2), p(n,3), ..., p(n,k)} can be partitioned into two sets with equal sums, where p(n,m) is m-th n-gonal number.

Original entry on oeis.org

4, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 3

Seiichi Manyama, Sep 17 2017

nonn

Conjecture: a(n) = 7 for n > 5.

			n = 3
1+3+6 = 10
n = 4
1+4+16+49 = 9+25+36 (= 70 = 28*4-42)
n = 5
1+5+22+35 = 12+51 (=63)
n = 6
1+6+28+91 = 15+45+66 (= 126 = 28*6-42)

Wikipedia, Polygonal number

Cf. A019568, A158092, A158380, A292474.

def f(k, n)
  n * ((k - 2) * n - k + 4) / 2
end
def A(k, n)
  ary = [1]
  s_ary = [0]
  (1..n).each{|i| s_ary << s_ary[-1] + f(k, i)}
  m = s_ary[-1]
  a = Array.new(m + 1){0}
  a[0] = 1
  (1..n).each{|i|
    b = a.clone
    (0..[s_ary[i - 1], m - f(k, i)].min).each{|j| b[j + f(k, i)] += a[j]}
    a = b
    s_ary[i] % 2 == 0 ? ary << a[s_ary[i] / 2] : ary << 0
  }
  ary
end
def B(n)
  i = 1
  while A(n, i)[-1] == 0
    i += 1
  end
  i
end
def A292510(n)
  (3..n).map{|i| B(i)}
end
p A292510(100)

p(n,1) + p(n,2) + p(n,4) + p(n,7) = p(n,3) + p(n,5) + p(n,6) (= 28*n-42). So a(n) <= 7.

cp's OEIS Frontend

A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

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A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n*(3*n-1)/2 = 0.

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A307877 Number of ways of partitioning the set of the first n positive squares into two subsets whose sums differ at most by 1.

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A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

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A350403 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 0 or 1.

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A369730 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 1.

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Maple

Mathematica

Formula

A292510 a(n) = smallest k >= 1 such that {1, p(n,2), p(n,3), ..., p(n,k)} can be partitioned into two sets with equal sums, where p(n,m) is m-th n-gonal number.

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Ruby

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A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n(3n-1)/2 = 0.