A231015 -id:A231015 - cp's OEIS frontend

A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 10, 0, 0, 86, 114, 0, 0, 478, 860, 0, 0, 5808, 10838, 0, 0, 55626, 100426, 0, 0, 696164, 1298600, 0, 0, 7826992, 14574366, 0, 0, 100061106, 187392994, 0, 0, 1223587084, 2322159814, 0, 0, 16019866270, 30353305134, 0, 0

Offset: 1

Pietro Majer, Mar 12 2009

nonn

Twice A083527.
Number of partitions of the half of the n-th-square-pyramidal number into parts that are distinct square numbers in the range 1 to n^2. Example: a(7)=2 since, squarePyramidal(7)=140 and 70=1+4+16+49=9+25+36. - Hieronymus Fischer, Oct 20 2010
Erdős & Surányi prove that this sequence is unbounded. More generally, there are infinitely many ways to write a given number k as such a sum. - Charles R Greathouse IV, Nov 05 2012
The expansion and integral representation formulas below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - Jonathan Sondow, Nov 11 2013

			For n=8 the a(8)=2 solutions are: +1-4-9+16-25+36+49-64=0 and -1+4+9-16+25-36-49+64=0.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.
P. Erdős and J. Surányi, Egy additív számelméleti probléma (in Hungarian; Russian and German summaries), Mat. Lapok 10 (1959), pp. 284-290.

Cf. A063865, A158118, A019568, A083527, A231015.

From Pietro Majer, Mar 15 2009: (Start)
N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*(x^(n^2)+x^(-n^2))):
a:=[op(a), coeff(p, x, 0)]: od:a; (End)
# second Maple program:
b:= proc(n, i) option remember; local m; m:= (1+(3+2*i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, 2*b(n^2, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 05 2012

b[n_, i_] := b[n, i] = With[{m = (1+(3+2*i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[ Abs[n-i^2], i-1] + b[n+i^2, i-1]]]]; a[n_] := If[Mod[n-1, 4]<2, 0, 2*b[n^2, n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(n)=2*sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012

from itertools import count, islice
from collections import Counter
def A158092_gen(): # generator of terms
    ccount = Counter({0:1})
    for i in count(1):
        bcount = Counter()
        for a in ccount:
            bcount[a+(j:=i**2)] += ccount[a]
            bcount[a-j] += ccount[a]
        ccount = bcount
        yield(ccount[0])
A158092_list = list(islice(A158092_gen(),20)) # Chai Wah Wu, Jan 29 2024

Constant term in the expansion of (x + 1/x)(x^4 + 1/x^4)..(x^n^2 + 1/x^n^2).
a(n)=0 for any n == 1 or 2 (mod 4).
Integral representation: a(n)=((2^n)/pi)*int_0^pi prod_{k=1}^n cos(x*k^2) dx
Asymptotic formula: a(n) = (2^n)*sqrt(10/(pi*n^5))*(1+o(1)) as n-->infty; n == -1 or 0 (mod 4).
a(n) = 2 * A083527(n). - T. D. Noe, Mar 12 2009
min{n : a(n) > 0} = A231015(0) = 7. - Jonathan Sondow, Nov 06 2013

a(51)-a(56) from R. H. Hardin, Mar 12 2009

Edited by N. J. A. Sloane, Sep 15 2009

A231071 Number of solutions to n = +- 1^2 +- 2^2 +- 3^2 +- 4^2 +- ... +- k^2 for minimal k giving at least one solution.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 9, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 2, 1, 2, 1, 14, 2, 1, 3, 2, 1, 2, 1, 1, 7, 1, 3, 2, 5, 1, 2, 1

Offset: 0

Alois P. Heinz, Nov 03 2013

This type of sequence was first studied by Andrica and Vacaretu. - Jonathan Sondow, Nov 06 2013

			a(8) = 3: 8 = -1-4-9-16+25-36+49 = -1-4+9+16-25-36+49 = -1+4+9-16+25+36-49.
a(9) = 2: 9 = -1-4+9+16+25-36 = 1+4+9-16-25+36.
a(10) = 1: 10 = -1+4-9+16.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dorin Andrica and Daniel Vacaretu, Representation theorems and almost unimodal sequences, Studia Univ. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

Cf. A000330, A231015, A231272.

Cf. A083527, A158092 (extremal sums).

b:= proc(n, i) option remember; (m->`if`(n>m, 0, `if`(n=m, 1,
      b(n+i^2, i-1) +b(abs(n-i^2), i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= proc(n) local k; for k while b(n, k)=0 do od; b(n, k) end:
seq(a(n), n=0..100);

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1,
   b[n+i^2, i-1] + b[Abs[n-i^2], i-1]]]][(1+(3+2*i)*i)*i/6];
a[n_] := Module[{k}, For[k = 1, b[n, k] == 0, k++]; b[n, k]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

From Jonathan Sondow, Nov 03 2013: (Start)
a(n(n+1)(2n+1)/6) = 1 for n > 0: n(n+1)(2n+1)/6 = 1+4+9+...+n^2. See A000330.
a(n(n+1)(2n+1)/6 - 2) = 1 for n > 1: n(n+1)(2n+1)/6 - 2 = -1+4+9+...+n^2. (End)

A231272 Numbers m with unique solution to m = +-1^2+-2^2+-3^2+-4^2+-...+-k^2 with minimal k giving at least one solution.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 35, 36, 37, 38, 44, 45, 47, 49, 51, 53, 55, 56, 57, 59, 60, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 76, 78, 81, 83, 86, 89, 91, 92, 94, 98, 100, 102, 106, 108, 109, 112

Offset: 1

Alois P. Heinz, Nov 06 2013

nonn

Numbers m such that A231071(m) = 1. The value of k is given by A231015(m).

			10 is member of the sequence with unique minimal solution 10 = -1+4-9+16.
A000330(k) = k(k+1)(2k+1)/6 = 1^2 + 2^2 + ... + k^2 is a member for k > 0. - _Jonathan Sondow_, Nov 06 2013

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Andrica, D., Vacaretu, D., Representation theorems and almost unimodal sequences, Studia Univ. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

Cf. A000330, A231015, A231071, A231016 (complement).

b:= proc(n, i) option remember; local m, t; m:= (1+(3+2*i)*i)*i/6;
      if n>m then 0 elif n=m then 1 else
         t:= b(abs(n-i^2), i-1);
         if t>1 then return 2 fi;
         t:= t+b(n+i^2, i-1); `if`(t>1, 2, t)
      fi
    end:
a:= proc(n) option remember; local m, k;
      for m from 1+ `if`(n=1, -1, a(n-1)) do
        for k while b(m, k)=0 do od;
        if b(m, k)=1 then return m fi
      od
    end:
seq(a(n), n=1..80);

b[n_, i_] := b[n, i] = Module[{m, t}, m = (1 + (3 + 2*i)*i)*i/6; If[n > m,  0, If[n == m, 1, t = b[Abs[n - i^2], i - 1]; If[t > 1, Return[2]]; t = t + b[n + i^2, i - 1]; If[t > 1, 2, t]]]];
a[n_] := a[n] = Module[{m, k}, For[m = 1 + If[n == 1, -1, a[n - 1]], True, m++, For[k = 1, b[m, k] == 0, k++]; If[b[m, k] == 1, Return[m]]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

A140358 Smallest nonnegative integer k such that n = +-1+-2+-...+-k for some choice of +'s and -'s.

Original entry on oeis.org

0, 1, 3, 2, 3, 5, 3, 5, 4, 5, 4, 5, 7, 5, 7, 5, 7, 6, 7, 6, 7, 6, 7, 9, 7, 9, 7, 9, 7, 9, 8, 9, 8, 9, 8, 9, 8, 9, 11, 9, 11, 9, 11, 9, 11, 9, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 15, 13, 15, 13

Offset: 0

John W. Layman, Jun 23 2008

			From _Seiichi Manyama_, Aug 18 2020: (Start)
Illustration of initial terms:
   0 =  0 (empty sum).
   1 =  1.
   2 =  1 - 2 + 3.
   3 =  1 + 2.
   4 = -1 + 2 + 3.
   5 =  1 + 2 + 3 + 4 - 5.
   6 =  1 + 2 + 3.
   7 =  1 + 2 + 3 - 4 + 5.
   8 = -1 + 2 + 3 + 4.
   9 =  1 + 2 - 3 + 4 + 5.
  10 =  1 + 2 + 3 + 4.
... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Rishi Advani, Formula for sequence on Mathematics Stack Exchange

Cf. A000217, A197702, A231015, A327467.

b:= proc(n,i) option remember;
      (n=0 and i=0) or n<=i*(i+1)/2 and (b(abs(n-i), i-1) or b(n+i, i-1))
    end:
a:= proc(n) local k;
      for k from 0 while not b(n,k) do od; k
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 19 2011

b[n_, i_] := b[n, i] = (n==0 && i==0) || Abs[n] <= i(i+1)/2 && (b[n-i, i-1] || b[n+i, i-1]);
a[n_] := Module[{k}, For[k = 0, !b[n, k], k++]; k];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 15 2020, after Alois P. Heinz *)

Conjecture when n is greater than 0. Choose k so that t(k)<=n

a(n) = a(-n) for all n in Z. - Seiichi Manyama, Aug 18 2020

Let k be the least integer such that t(k) >= n. If t(k) and n have the same parity then a(n) = k. Otherwise a(n) is equal to the least odd integer greater than k. - Rishi Advani, Jan 24 2021

a(0)=3 prepended by Seiichi Manyama, Aug 17 2020

Edited and a(0)=0 from Alois P. Heinz, Aug 18 2020

A231016 Numbers m with non-unique solution to m = +- 1^2 +- 2^2 +- ... +- k^2 with minimal k giving at least one solution.

Original entry on oeis.org

0, 8, 9, 16, 18, 25, 31, 32, 33, 34, 39, 40, 41, 42, 43, 46, 48, 50, 52, 54, 58, 61, 67, 69, 74, 75, 77, 79, 80, 82, 84, 85, 87, 88, 90, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 110, 111, 113, 115, 116, 117, 118, 121, 123, 127, 129, 131, 133, 135, 137, 141

Offset: 1

Jonathan Sondow, Nov 06 2013

nonn

The minimal k = A231015(m).

Complement of A231272.

			0 = 1 + 4 - 9 + 16 - 25 - 36 + 49 = sum with signs reversed, so 0 is a member.
9 = - 1 - 4 + 9 + 16 + 25 - 36 = 1 + 4 + 9 - 16 - 25 + 36, so 9 is a member.
A000330(k) = k(k+1)(2k+1)/6 = 1^2 + 2^2 + ... + k^2 is not a member, for k > 0.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Andrica, D., Vacaretu, D., Representation theorems and almost unimodal sequences, Studia Univ. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

Cf. A000330, A231015, A231071, A231272.

b:= proc(n, i) option remember; local m, t; m:= (1+(3+2*i)*i)*i/6;
      if n>m then 0 elif n=m then 1 else
         t:= b(abs(n-i^2), i-1);
         if t>1 then return 2 fi;
         t:= t+b(n+i^2, i-1); `if`(t>1, 2, t)
      fi
    end:
a:= proc(n) option remember; local m, k;
      for m from 1+ `if`(n=1, -1, a(n-1)) do
        for k while b(m, k)=0 do od;
        if b(m, k)>1 then return m fi
      od
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 06 2013

b[n_, i_] := b[n, i] = Module[{m, t}, m = (1+(3+2*i)*i)*i/6; Which[n>m, 0, n == m, 1, True, t = b[Abs[n-i^2], i-1]; If[t>1, Return[2]]; t = t + b[n+i^2, i-1]; If[t>1, 2, t]]]; a[n_] := a[n] = Module[{m, k}, For[m = 1 + If[n == 1, -1, a[n-1]], True, m++, For[k = 1, b[m, k] == 0, k++]; If[b[m, k]>1, Return[m]]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 28 2014, after Alois P. Heinz *)

{ n : A231071(n) > 1 }.

Showing 1-5 of 5 results.

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A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

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A231071 Number of solutions to n = +- 1^2 +- 2^2 +- 3^2 +- 4^2 +- ... +- k^2 for minimal k giving at least one solution.

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A231272 Numbers m with unique solution to m = +-1^2+-2^2+-3^2+-4^2+-...+-k^2 with minimal k giving at least one solution.

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A140358 Smallest nonnegative integer k such that n = +-1+-2+-...+-k for some choice of +'s and -'s.

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A231016 Numbers m with non-unique solution to m = +- 1^2 +- 2^2 +- ... +- k^2 with minimal k giving at least one solution.

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