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

%I

%S 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,

%T 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,

%U 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

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

%C This type of sequence was first studied by Andrica and Vacaretu. - _Jonathan Sondow_, Nov 06 2013

%H Alois P. Heinz, <a href="/A231071/b231071.txt">Table of n, a(n) for n = 0..10000</a>

%H Andrica, D., Vacaretu, D., <a href="http://www.cs.ubbcluj.ro/~studia-m/2006-4/andrica.pdf">Representation theorems and almost unimodal sequences</a>, Studia Univ. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

%F From _Jonathan Sondow_, Nov 03 2013: (Start)

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

%F 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)

%e 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.

%e a(9) = 2: 9 = -1-4+9+16+25-36 = 1+4+9-16-25+36.

%e a(10) = 1: 10 = -1+4-9+16.

%p b:= proc(n, i) option remember; (m->`if`(n>m, 0, `if`(n=m, 1,

%p b(n+i^2, i-1) +b(abs(n-i^2), i-1))))((1+(3+2*i)*i)*i/6)

%p end:

%p a:= proc(n) local k; for k while b(n, k)=0 do od; b(n, k) end:

%p seq(a(n), n=0..100);

%Y Cf. A000330, A231015, A231272.

%Y Cf. A083527, A158092 (extremal sums).

%K nonn,look

%O 0,1

%A _Alois P. Heinz_, Nov 03 2013

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Last modified July 29 22:13 EDT 2021. Contains 346346 sequences. (Running on oeis4.)