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A231015 Least k such that n = +- 1^2 +- 2^2 +- 3^2 +- 4^2 +- ... +- k^2 for some choice of +- signs. 5
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 (list; graph; refs; listen; history; text; internal format)
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 Andrica-Ionascu links for more comments.

The integral formula (3.6) in Andrica-Vacaretu (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)). Andrica-Vacaretu used this generating function to prove the integral formula. Section 4 of Andrica-Vacaretu 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. Babes-Bolyai, Mathematica, Vol. LI, 4 (2006), 23-33.

P. Erdős and J. Surányi, Egy additív számelméleti probléma (in Hungarian; Russian and German summaries), Mat. Lapok 10 (1959), pp. 284-290.

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, i-1) or b(abs(n-i^2), i-1))

    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, i-1] || b[Abs[n-i^2], i-1])]; a[n_] := Module[{k}, For[k = 1, !b[n, k] , k++]; k]; Table[a[n], {n, 0, 100}] (* Jean-François 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

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Last modified May 25 03:14 EDT 2019. Contains 323539 sequences. (Running on oeis4.)