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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.
4
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 06 2013
STATUS
approved