cp's OEIS Frontend

A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.

%I A051132 #29 Jul 02 2025 16:01:58
%S A051132 0,1,9,25,45,69,109,145,193,249,305,373,437,517,609,697,793,889,1005,
%T A051132 1125,1245,1369,1513,1649,1789,1941,2109,2285,2449,2617,2809,2997,
%U A051132 3205,3405,3613,3841,4049,4281,4509,4765,5013,5249,5521,5785,6073,6349,6621
%N A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.
%H A051132 T. D. Noe, <a href="/A051132/b051132.txt">Table of n, a(n) for n = 0..1000</a>
%H A051132 Jianqiang Zhao, <a href="https://arxiv.org/abs/2505.06234">The Largest Circle Enclosing n Lattice Points</a>, arXiv:2505.06234 [math.GM], 2025. See p. 18.
%F A051132 a(n) = A000328(n) - A046109(n). - _Reinhard Zumkeller_, Jan 23 2012
%F A051132 Limit_{n->oo} a(n)/n^2 = Pi. - _Chai Wah Wu_, Feb 12 2025
%F A051132 a(n) = 4*n - 3 + 4 Sum_{i=1..n-1} ceiling(sqrt(n^2 - i^2)) - 1, for n > 0 (see Zhao). - _Giorgos Kalogeropoulos_, May 20 2025
%e A051132 a(3)=25 from the points of shapes 00 (1), 10 (4), 11 (4), 20 (4), 21 (8), 22 (4).
%t A051132 Table[Sum[SquaresR[2, k], {k, 0, n^2 - 1}], {n, 0, 46}]
%t A051132 a[0]=0;a[n_]:=4*n-3+4Sum[Ceiling[Sqrt[n^2-i^2]]-1,{i,n-1}];Array[a,47,0] (* _Giorgos Kalogeropoulos_, May 20 2025 *)
%o A051132 (Haskell)
%o A051132 a051132 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 < n^2]
%o A051132 -- _Reinhard Zumkeller_, Jan 23 2012
%o A051132 (Python)
%o A051132 from math import isqrt
%o A051132 def A051132(n): return 1+(sum(isqrt(k*((n<<1)-k)-1) for k in range(1,n+1))<<2) if n else 0 # _Chai Wah Wu_, Feb 12 2025
%Y A051132 Changing "<" to "<=" in the definition gives A000328.
%K A051132 nonn,nice,easy
%O A051132 0,3
%A A051132 Jostein Trondal (jostein.trondal(AT)protech.no)
%E A051132 More terms from _James Sellers_