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.

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

Author

Alois P. Heinz, Nov 06 2013

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

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

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);

Mathematica

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 *)