A231071 Number of solutions to n = +- 1^2 +- 2^2 +- 3^2 +- 4^2 +- ... +- k^2 for minimal k giving at least one solution.

2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 9, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 2, 1, 2, 1, 14, 2, 1, 3, 2, 1, 2, 1, 1, 7, 1, 3, 2, 5, 1, 2, 1

Offset

0,1

Comments

This type of sequence was first studied by Andrica and Vacaretu. - Jonathan Sondow, Nov 06 2013

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Dorin Andrica and Daniel Vacaretu, Representation theorems and almost unimodal sequences, Studia Univ. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

Formula

From Jonathan Sondow, Nov 03 2013: (Start)

a(n(n+1)(2n+1)/6) = 1 for n > 0: n(n+1)(2n+1)/6 = 1+4+9+...+n^2. See A000330.

a(n(n+1)(2n+1)/6 - 2) = 1 for n > 1: n(n+1)(2n+1)/6 - 2 = -1+4+9+...+n^2. (End)

Example

a(8) = 3: 8 = -1-4-9-16+25-36+49 = -1-4+9+16-25-36+49 = -1+4+9-16+25+36-49.
a(9) = 2: 9 = -1-4+9+16+25-36 = 1+4+9-16-25+36.
a(10) = 1: 10 = -1+4-9+16.

Maple

b:= proc(n, i) option remember; (m->`if`(n>m, 0, `if`(n=m, 1,
      b(n+i^2, i-1) +b(abs(n-i^2), i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= proc(n) local k; for k while b(n, k)=0 do od; b(n, k) end:
seq(a(n), n=0..100);

Mathematica

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1,
   b[n+i^2, i-1] + b[Abs[n-i^2], i-1]]]][(1+(3+2*i)*i)*i/6];
a[n_] := Module[{k}, For[k = 1, b[n, k] == 0, k++]; b[n, k]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000330, A231015, A231272.

Cf. A083527, A158092 (extremal sums).

Sequence in context: A186027 A359250 A322482 * A209156 A329325 A341149

Adjacent sequences: A231068 A231069 A231070 * A231072 A231073 A231074

Keyword

nonn,look

Author

Alois P. Heinz, Nov 03 2013

Status

approved