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A160663 Number of distinct sums that one can obtain by adding two squares among the n first ones. 4
2, 5, 9, 14, 19, 26, 33, 41, 50, 60, 70, 82, 93, 105, 119, 134, 147, 164, 179, 197, 215, 234, 251, 272, 293, 314, 336, 359, 381, 407, 430, 456, 483, 507, 535, 566, 594, 623, 652, 686, 714, 748, 780, 812, 849, 883, 918, 956, 992, 1030, 1068, 1107, 1141, 1181 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Essentially the same as A047800: a(n) = A047800(n) - 1.

Let A be the set of the n first squares (1,4,9,...,n^2). Let A+A be the corresponding sumset (= {a,b,a+b where (a,b) in A^2}). That very sequence describes the number of elements of A+A, relatively to n.

a(n-1) is the number of distinct positive distances on an n X n pegboard. What is its asymptotic growth? Can it be efficiently computed for large n? - Charles R Greathouse IV, Jun 13 2013

REFERENCES

Melvyn B. Nathanson (1996). "Additive Number Theory: the Classical Bases" Graduate Texts in Mathematics. 164. Springer-Verlag. p. 192. ISBN 0-387-94656-X.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690.

L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690. doi:10.1007/BF01448914.

Samuel S. Wagstaff, Jr., The Schnirelmann Density of the Sums of Three Squares, Proc. Amer. Math. Soc. 52 (1975), 1-7.

Wikipedia, Additive number theory

Wikipedia, Schnirelmann density

Wikipedia, Edmund Landau

FORMULA

a(n) = card(A+A) where A={k^2} k=1..n and A+A = {a,b,a+b where (a,b) in A^2}.

Trivially 2n <= a(n) <= n(n+1)/2. - Charles R Greathouse IV, Oct 30 2015

a(n) << n^2/sqrt(log n). - Charles R Greathouse IV, Oct 30 2015

EXAMPLE

For n = 3, A = {1,4,9}, A+A = {1,4,9} U {2,5,10,8,13,18} thus A+A = {1,2,4,5,8,9,10,13,18}, and hence card(A+A) = 9; a(3) = 9.

MAPLE

a:= proc(n) local A, i, j; A:= [i^2$i=1..n]; nops([{A[], seq (seq (A[i]+A[j], j=1..i), i=1..nops(A))}[]]) end: seq (a(n), n=1..60); # Alois P. Heinz, Jun 16 2009

MATHEMATICA

a[n_] := (Table[i^2 + j^2, {i, 0, n}, {j, i, n}] // Flatten // Union // Length) - 1; Array[a, 60] (* Jean-François Alcover, May 25 2018 *)

PROG

(Python)

def a(n):

....SUM, SQR = set(), set([x ** 2 for x in range(1, n + 1)])

....for i in SQR:

........SUM.add(i)

........for j in SQR: SUM.add(i + j)

....return len(SUM)

# Romain CARRE (romain.carre.2008(AT)enseirb.fr), Apr 16 2010

(PARI) a(n)=#vecsort(vector(n^2, i, ((i-1)\n)^2+((i-1)%n)^2), , 8)-1 \\ Charles R Greathouse IV, Jun 13 2013

CROSSREFS

Sequence in context: A266899 A112265 A025281 * A024201 A110443 A130029

Adjacent sequences:  A160660 A160661 A160662 * A160664 A160665 A160666

KEYWORD

nonn

AUTHOR

Romain CARRE (romain.carre.2008(AT)enseirb.fr), May 22 2009

EXTENSIONS

More terms from Alois P. Heinz, Jun 16 2009

STATUS

approved

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Last modified November 21 04:47 EST 2019. Contains 329350 sequences. (Running on oeis4.)