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

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

Author

Jonathan Sondow, Nov 06 2013

Keywords

• nonn

Comments

The minimal k = A231015(m).

Complement of A231272.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000330, A231015, A231071, A231272.

Programs

Maple

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

Mathematica

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 *)

Formula

{ n : A231071(n) > 1 }.