A047800 Number of different values of i^2 + j^2 for i,j in [0, n].

1, 3, 6, 10, 15, 20, 27, 34, 42, 51, 61, 71, 83, 94, 106, 120, 135, 148, 165, 180, 198, 216, 235, 252, 273, 294, 315, 337, 360, 382, 408, 431, 457, 484, 508, 536, 567, 595, 624, 653, 687, 715, 749, 781, 813, 850, 884, 919, 957, 993, 1031, 1069, 1108, 1142

Offset

0,2

Comments

a(n-1) is the number of distinct 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

Conjecture (after Landau and Erdős): a(n) ~ c * n^2 / sqrt(log(n)), where c = 0.79... . - Vaclav Kotesovec, Mar 10 2016

Links

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from T. D. Noe)

Erdős, P., On sets of distances of n points, American Mathematical Monthly 53, pp. 248-250 (1946).

Vaclav Kotesovec, Graph - The asymptotic ratio

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig B. G. Teubner, 1909, p. 643.

Mathematica

Table[ Length@Union[ Flatten[ Table[ i^2+j^2, {i, 0, n}, {j, 0, n} ] ] ], {n, 0, 49} ]
nmax = 100; sq = Table[i^2 + j^2, {i, 0, nmax}, {j, 0, nmax}]; Table[Length@Union[Flatten[Table[Take[sq[[j]], n + 1], {j, 1, n + 1}]]], {n, 0, nmax}] (* Vaclav Kotesovec, Mar 09 2016 *)

Prog

(Haskell)
import Data.List (nub)
a047800 n = length $ nub [i^2 + j^2 | i <- [0..n], j <- [i..n]]
-- Reinhard Zumkeller, Oct 03 2012
(PARI) a(n)=#vecsort(vector(n^2, i, ((i-1)\n)^2+((i-1)%n)^2), , 8) \\ Charles R Greathouse IV, Jun 13 2013

Crossrefs

Cf. A034966, A047801, A160663.

Sequence in context: A027920 A033438 A037452 * A109443 A138777 A096895

Adjacent sequences: A047797 A047798 A047799 * A047801 A047802 A047803

Keyword

nonn,easy,nice

Author

Wouter Meeussen

Status

approved