A231016 Numbers n with non-unique solution to n = +- 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

Offset

1,2

Comments

The minimal k = A231015(n).

Complement of A231272.

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.

Formula

{ n : A231071(n) > 1 }.

Example

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.

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 *)

Crossrefs

Cf. A000330, A231015, A231071, A231272.

Sequence in context: A057111 A171425 A143720 * A346090 A169817 A351098

Adjacent sequences: A231013 A231014 A231015 * A231017 A231018 A231019

Keyword

nonn

Author

Jonathan Sondow, Nov 06 2013

Status

approved