

A231015


Least k such that n = + 1^2 + 2^2 + 3^2 + 4^2 + ... + k^2 for some choice of + signs.


6



7, 1, 4, 2, 3, 2, 3, 6, 7, 6, 4, 6, 3, 5, 3, 5, 7, 6, 7, 6, 4, 5, 4, 5, 7, 9, 7, 5, 4, 5, 4, 6, 7, 6, 7, 5, 7, 5, 7, 6, 7, 6, 7, 9, 8, 5, 8, 5, 7, 6, 7, 6, 11, 5, 8, 5, 7, 6, 7, 6, 7, 10, 7, 6, 7, 6, 7, 9, 7, 10, 7, 6, 7, 6, 8, 9, 8, 9, 8, 9, 7, 6, 7, 6, 11, 9
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OFFSET

0,1


COMMENTS

Erdős and Surányi proved that for each n there are infinitely many k satisfying the equation.
A158092(k) is the number of solutions to 0 = +1^2 + 2^2 + ... + k^2. The first nonzero value is A158092(7) = 2, so a(0) = 7.
a(n) is also defined for n < 0, and clearly a(n) = a(n).
See A158092 and the AndricaIonascu links for more comments.
The integral formula (3.6) in AndricaVacaretu (see Theorem 3 of the INTEGERS 2013 slides which has a typo) gives in this case the number of representations of n as + 1^2 + 2^2 + ... + k^2 for some choice of + signs. This integral formula is (2^n/2*Pi)*int _0^{2*Pi} cos(n*t) * prod_{j=1..k} cos(j^2*t) dt. Clearly the number of such representations of n is the coefficient of z^n in the expansion (z^(1^2)+z^(1^2))*(z^(2^2)+z^(2^2))*..*(z^(k^2)+z^(k^2)). AndricaVacaretu used this generating function to prove the integral formula. Section 4 of AndricaVacaretu gives a table of the number of such representations of n for k=1,..,9.  Dorin Andrica, Nov 12 2013


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, Integers Conference 2013 Abstract.
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
D. Andrica and D. Vacaretu, Representation theorems and almost unimodal sequences, Studia Univ. BabesBolyai, Mathematica, Vol. LI, 4 (2006), 2333.
P. Erdős and J. Surányi, Egy additív számelméleti probléma (in Hungarian; Russian and German summaries), Mat. Lapok 10 (1959), pp. 284290.


FORMULA

a(n(n+1)(2n+1)/6) = a(A000330(n)) = n for n > 0.
a((n(n+1)(2n+1)/6)2) = a(A000330(n)2) = n for n > 0.


EXAMPLE

0 = 1^2 + 2^2  3^2 + 4^2  5^2  6^2 + 7^2.
1 = 1^2.
2 =  1^2  2^2  3^2 + 4^2.
3 =  1^2 + 2^2.
4 =  1^2  2^2 + 3^2.


MAPLE

b:= proc(n, i) option remember; local m; m:=i*(i+1)*(2*i+1)/6;
n<=m and (n=m or b(n+i^2, i1) or b(abs(ni^2), i1))
end:
a:= proc(n) local k; for k while not b(n, k) do od; k end:
seq(a(n), n=0..100); # Alois P. Heinz, Nov 03 2013


MATHEMATICA

b[n_, i_] := b[n, i] = Module[{m}, m = i*(i+1)*(2*i+1)/6; n <= m && (n == m  b[n+i^2, i1]  b[Abs[ni^2], i1])]; a[n_] := Module[{k}, For[k = 1, !b[n, k] , k++]; k]; Table[a[n], {n, 0, 100}] (* JeanFrançois Alcover, Jan 28 2014, after Alois P. Heinz *)


CROSSREFS

Cf. A000330, A158092, A231071, A231272.
Sequence in context: A010504 A242132 A011450 * A183031 A021018 A010145
Adjacent sequences: A231012 A231013 A231014 * A231016 A231017 A231018


KEYWORD

nonn,look


AUTHOR

Jonathan Sondow, Nov 02 2013


EXTENSIONS

a(4) corrected and a(5)a(85) from Donovan Johnson, Nov 03 2013


STATUS

approved



