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 A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0). 31
 1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Michael Gilleland, Some Self-Similar Integer Sequences Eric Weisstein's World of Mathematics, Circle Lattice Points FORMULA a(n) = A000328(n) - A051132(n). a(n) = 8*A046080(n) + 4 for n > 0. a(n) = A004018(n^2). From Jean-Christophe Hervé, Dec 01 2013: (Start) a(A084647(k)) = 28. a(A084648(k)) = 36. a(A084649(k)) = 44. (End) a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017 a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018 From Hugo Pfoertner, Sep 21 2023: (Start) a(n) = 8*A063014(n) - 4 for n > 0. a(n) = 4*A256452(n) for n > 0. (End) EXAMPLE a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5. MAPLE N:= 1000: # to get a(0) to a(N) A:= Array(0..N): A[0]:= 1: for x from 1 to N do A[x]:= A[x]+4; for y from 1 to min(x-1, floor(sqrt(N^2-x^2))) do z:= x^2+y^2; if issqr(z) then t:= sqrt(z); A[t]:= A[t]+8; fi od od: seq(A[i], i=0..N); # Robert Israel, May 08 2015 MATHEMATICA Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *) PROG (Haskell) a046109 n = length [(x, y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2] -- Reinhard Zumkeller, Jan 23 2012 (Python) from sympy import factorint def a(n): r = 1 for p, e in factorint(n).items(): if p%4 == 1: r *= 2*e + 1 return 4*r if n > 0 else 0 # Orson R. L. Peters, Jan 31 2017 (PARI) a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017 (PARI) a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017 CROSSREFS Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452. Cf. A000328, A051132. Sequence in context: A295643 A190718 A035621 * A294246 A107680 A358509 Adjacent sequences: A046106 A046107 A046108 * A046110 A046111 A046112 KEYWORD nonn,easy,nice AUTHOR Eric W. Weisstein STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)