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A231016 Numbers n with non-unique solution to n = +- 1^2 +- 2^2 +- ... +- k^2 with minimal k giving at least one solution. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 28 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A000330, A231015, A231071, A231272.

Sequence in context: A057111 A171425 A143720 * A169817 A227649 A227648

Adjacent sequences:  A231013 A231014 A231015 * A231017 A231018 A231019

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Nov 06 2013

STATUS

approved

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Last modified May 21 08:48 EDT 2019. Contains 323441 sequences. (Running on oeis4.)