A231272 Numbers n with unique solution to n = +-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

Offset

1,2

Comments

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

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.

Example

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

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 *)

Crossrefs

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

Sequence in context: A031492 A350076 A035060 * A143719 A078779 A351958

Adjacent sequences: A231269 A231270 A231271 * A231273 A231274 A231275

Keyword

nonn

Author

Alois P. Heinz, Nov 06 2013

Status

approved