A046109 -id:A046109 - cp's OEIS frontend

A088899 T(n, k) = number of ordered pairs of integers (x,y) with x^2/n^2 + y^2/k^2 = 1, 1 <= k <= n; triangular array, read by rows.

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Oct 21 2003

T(n,k) is the number of lattice points on the circumference of an ellipse with semimajor axis = n, semiminor axis = k and center = (0,0).

			From _Antti Karttunen_, Nov 08 2018: (Start)
Triangle begins:
---------------------------------------------------------------
k=    1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
---------------------------------------------------------------
n= 1: 4;
n= 2: 4,  4;
n= 3: 4,  4,  4;
n= 4: 4,  4,  4,  4;
n= 5: 4,  4,  4,  4, 12;
n= 6: 4,  4,  4,  4,  4,  4;
n= 7: 4,  4,  4,  4,  4,  4,  4;
n= 8: 4,  4,  4,  4,  4,  4,  4,  4;
n= 9: 4,  4,  4,  4,  4,  4,  4,  4,  4;
n=10: 4,  4,  4,  4, 12,  4,  4,  4,  4, 12;
n=11: 4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4;
n=12: 4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4;
n=13: 4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4, 12;
n=14: 4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4;
n=15: 4,  4,  4,  4, 12,  4,  4,  4,  4, 12,  4,  4,  4,  4, 12;
---
T(5,5) = 12 as there are following 12 solutions for pair (5,5): (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4), (4, -3).
T(15,10) = 12, as there are following 12 solutions for pair (15,10): (-15,0), (-12,-6), (-12,6), (-9,-8), (-9,8), (0,-10), (0,10), (9,-8), (9,8), (12,-6), (12,6), (15,0).
(End)

Antti Karttunen, Rows n = 1..225 of triangle, flattened
Eric Weisstein's World of Mathematics, Ellipse

Cf. A046109, A088897, A088898.

T[n_, k_] := Reduce[x^2/n^2 + y^2/k^2 == 1, {x, y}, Integers] // Length;
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 27 2021 *)

up_to = 105;
A088899tr(n,k) = { my(s=0, t=(n^2)*(k^2)); for(x=-n,n,for(y=-k,k,if((x*x*k*k)+(y*y*n*n) == t, s++))); (s); };
A088899list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, if(i++ > up_to, return(v)); v[i] = A088899tr(n,k))); (v); };
v088899 = A088899list(up_to);
A088899(n) = v088899[n]; \\ Antti Karttunen, Nov 07 2018

a(n) = A088897(n) - A088898(n);

T(n,n) = A046109(n).

A292276 a(n) = number of vertices of the convex hull of the set of points of norm <= n^2 in square lattice.

Original entry on oeis.org

1, 4, 4, 8, 12, 12, 12, 12, 20, 12, 20, 20, 20, 20, 20, 20, 20, 24, 28, 20, 20, 20, 36, 36, 28, 20, 36, 36, 36, 36, 28, 36, 36, 36, 44, 36, 36, 36, 36, 52, 44, 36, 36, 36, 52, 44, 52, 52, 44, 52, 44, 60, 52, 44, 52, 44, 52, 52, 52, 52, 60, 52, 52, 52, 68, 44

Offset: 0

Rémy Sigrist, Sep 13 2017

nonn

The convex hull of a finite point set in dimension 2, say S, forms a convex polygon whose vertices are in S.
For any n >= 0, A000328(n) gives the number of elements of the set of points of norm <= n^2 in square lattice.
For symmetry reasons, a(n) is a multiple of 4 for any n > 0.

			See Links section.

Rémy Sigrist, C++ program for A292276
Rémy Sigrist, Illustration of the convex hulls for n=0..10
Wikipedia, Convex hull of a finite point set

Cf. A000328, A046109.

A046110 Number of lattice points on circumference of a circle of radius (2n+1)/2 with center at (1/2,0).

Original entry on oeis.org

2, 2, 6, 2, 2, 2, 6, 6, 6, 2, 2, 2, 10, 2, 6, 2, 2, 6, 6, 6, 6, 2, 6, 2, 2, 6, 6, 6, 2, 2, 6, 2, 18, 2, 2, 2, 6, 10, 2, 2, 2, 2, 18, 6, 6, 6, 2, 6, 6, 2, 6, 2, 6, 2, 6, 6, 6, 6, 6, 6, 2, 6, 14, 2, 2, 2, 2, 6, 6, 2, 2, 6, 18, 2, 6, 2, 6, 6, 6, 6, 2, 2, 6, 2, 10, 2, 6, 10, 2, 2, 6, 6, 18, 6, 2, 2, 6, 18

Offset: 0

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Circle Lattice Points.

Cf. A046109.

a(n) = 4 * A046080(2*n+1) + 2. - Sean A. Irvine, Apr 02 2021

A046111 Number of lattice points on circumference of a circle of radius 1/3,2/3,4/3,5/3,... with center at (1/3,0).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 1, 3, 3, 3, 1, 3, 3, 1, 1, 1, 1, 1, 5, 3, 3, 3, 1, 3, 1, 3, 1, 1, 9, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 3, 1, 9, 1, 1, 3, 3, 1, 1, 3, 3, 1, 5, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 7, 1, 1, 9, 1, 1, 1, 3, 3, 1, 3, 1, 3, 9, 3, 3, 3, 1, 1

Offset: 0

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Circle Lattice Points.

Cf. A046109.

A256452 Number of integer solutions to n^2 = x^2 + y^2 with x>0, y>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 1, 1, 1, 1, 5, 3, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 1, 9, 1, 1, 3, 1, 3, 1, 1, 3, 3, 5, 1, 1, 3, 1, 3, 1, 3, 1, 1, 9, 1, 3

Offset: 1

Michael Somos, Mar 29 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046080, A046109, A002654.

a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n^2)): seq(a(n), n=1..100);  # Ridouane Oudra, Aug 18 2024

a[ n_] := Sum[ Mod[ Length@Divisors[n^2 - k^2], 2], {k, n}];
a[ n_] := Length @ FindInstance[ n^2 == x^2 + y^2 && x > 0 && y >= 0, {x, y}, Integers, 10^9]; (* Michael Somos, Aug 15 2016 *)
f[p_, e_] := If[Mod[p, 4] == 1, 2*e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)

{a(n) = sum(k=1, n, issquare(n^2 - k^2))};

Multiplicative with a(p^e) = 2*e + 1 if p == 1 (mod 4), otherwise a(p^e) = 1.
a(n) = 1 + 2*A046080(n) if n>0.
a(n) = A046109(n)/4 for n > 0. - Hugo Pfoertner, Sep 21 2023
a(n) = A002654(n^2). - Ridouane Oudra, Aug 18 2024

A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

Original entry on oeis.org

1, 5, 9, 13, 17, 29, 33, 37, 41, 45, 57, 61, 65, 77, 81, 93, 97, 109, 113, 117, 129, 133, 137, 141, 145, 165, 177, 181, 185, 197, 209, 213, 217, 221, 233, 245, 249, 261, 265, 277, 289, 301, 305, 309, 313, 325, 329, 333, 337, 341, 361, 373, 385, 397, 401, 413, 417, 421, 433, 437, 449

Offset: 0

Alexander Kritov, Nov 21 2021

nonn

Consider a 2D lattice, where the Cartesian coordinates x and y are legs of the Pythagorean triangle. Thus the notion of Pythagorean triple is extended to the cases when sides x, y are in Z (i.e., sides also include negative integers and zero). The sequence gives the number of such triples on or inside a circle of radius n.

Partial sums of A046109.

			Sides (coordinates)                                                       a(n)
------------------------------------------------------------------------------
(0,0)                                                                       1
(-1,0)(0,-1)(0,1)(1,0)                                                      5
(-2,0)(0,-2)(0,2)(2,0)                                                      9
(-3,0)(0,-3)(0,3)(3,0)                                                     13
(-4,0)(0,-4)(0,4)(4,0)                                                     17
(-5,0)(-4,-3)(-4,3)(-3,-4)(-3,4)(0,-5)(0,5)(3,-4)(3,4)(4,-3)(4,3)(5,0)     29
(-6,0)(0,-6)(0,6)(6,0)                                                     33
(-7,0)(0,-7)(0,7)(7,0)                                                     37
(-8,0)(0,-8)(0,8)(8,0)                                                     41
(-9,0)(0,-9)(0,9)(9,0)                                                     45
(-10,0)(-8,-6)(-8,6)(-6,-8)(-6,8)(0,-10)(0,10)(6,-8)(6,8)(8,-6)(8,6)(10,0) 57
(-11,0)(0,-11)(0,11)(11,0)                                                 61
(-12,0)(0,-12)(0,12)(12,0)                                                 65

Alexander Kritov, Source code

Cf. A046080, A211432, A046109 (first differences), A349536 (in 1/8 sector).

C
```
/* See links */
```

f(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)); \\ A046109
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, Nov 27 2021

a(n) = (A211432(n) + 1)/2.

a(n) = a(n-1) + 4 + 8*A046080(n).

A355760 a(n) is the number of grid points in a square lattice covered by the area enclosed by n loops of an Archimedean spiral with starting point (0, 0) and endpoint (n, 0).

Original entry on oeis.org

1, 2, 8, 21, 40, 64, 97, 132, 178, 228, 282, 350, 415, 492, 574, 660, 756, 855, 962, 1076, 1195, 1322, 1451, 1590, 1736, 1885, 2044, 2204, 2378, 2552, 2734, 2922, 3116, 3317, 3525, 3741, 3960, 4187, 4416, 4655, 4900, 5154, 5410, 5674, 5946, 6223, 6502, 6791, 7087, 7391, 7698

Offset: 0

Karl-Heinz Hofmann, Jul 16 2022

nonn

Grid points coincident with the outer boundary of the spiral are included.
The spiral figure is closed with a line from (n, 0) to (n-1, 0).
Conjecture: Only lattice points on the positive x-axis are on the outer boundary of the spiral. It seems that the spiral passes all other grid points without hitting any.

			See the PDF in links.

Karl-Heinz Hofmann, Examples of a(1..8)
Wikipedia, Archimedean spiral

Cf. A000328, A295344, A046109.

A350459 Number of positive rational points on the unit circle with denominator <= n and numerator >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 4, 4, 5, 5, 10, 10, 11, 11, 11, 19, 19, 19, 19, 19, 29, 32, 32, 32, 33, 44, 44, 44, 44, 47, 60, 60, 61, 61, 66, 82, 83, 83, 83, 83, 100, 100, 100, 100, 100, 122, 127, 134, 135, 135, 156, 156, 156, 159, 159, 183, 184, 184, 184, 184, 220, 220, 220, 227, 227, 254

Offset: 1

Benoit Cloitre, Jan 01 2022

nonn

A rational point (x,y) is of the form (a/c, b/d) with (a,b,c,d) integers. Sequence gives the number of quadruples (a,b,c,d) satisfying a >= b >= 1, 1 <= c <= n, 1 <= d <= n and such that a^2/c^2 + b^2/d^2 = 1.

Cf. A046109.

a(n)=sum(a=1,n,sum(b=1,a,sum(c=1,n,sum(d=1,n,if(a^2/c^2+b^2/d^2-1,0,1)))))

def A350459(n): return sum(1 for d in range(1,n+1) for c in range(1,n+1) for b in range(1,d+1) for a in range(1, b+1) if (a*d)**2 + (b*c)**2 == (c*d)**2) # Chai Wah Wu, Jan 19 2022

A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.

Original entry on oeis.org

0, 16, 64, 144, 256, 912, 576, 784, 1024, 1296, 3648, 1936, 2304, 7312, 3136, 8208, 4096, 11824, 5184, 5776, 14592, 7056, 7744, 8464, 9216, 41232, 29248, 11664, 12544, 27568, 32832, 15376, 16384, 17424, 47296, 44688, 20736, 61104, 23104, 65808, 58368, 78096, 28224, 29584, 30976, 73872

Offset: 0

Torlach Rush, Aug 08 2019

nonn

For this sequence the square spiral begins with 0 and is the second illustration in the comments of A317186, where 0 is the origin of our circles.
a(n) >= A001107(n) + A033991(n) + A007742(n) + A033954(n).
a(n) = A016802(n) iff A046109(n) = 4.
a(n) = A016802(n) iff n <> k * A002144(m), k,m >= 1.
a(n) is congruent to 0 mod 16 and is the sum of one or more terms of A016802.
Conjecture: a(n) is a term of A277699 iff a(n)/16 = A277699(n).

			16 is a term because 16 = 16*(1)^2.
912 is a term because 912 = 16*(5)^2 + (2*(16*(4)^2)).
41232 is a term because 41232 = 16*(25)^2 + (2*((16*(24)^2) + (16*(20)^2))).

Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A001107, A002144, A007742, A016802, A033954, A033991, A046109, A277699, A317186.

Tb(n) = {return(16 * n * n)}
llsum(n) = {my(x=0); for (i = 1, n - 2, for (ii = i+1, n - 1, if(n*n == (ii*ii) + (i*i), x+=(2 * Tb(ii))))); return(x)}
Tx(n) = {my(x=0); forprimestep(x = 5, n, 4, if(n%x==0, return(llsum(n))))}
Tn(n) = {for (i = 0, n, print1(Tb(i) + Tx(i), ", "))}
Tn(45)

A365620 Number of integer grid points on the circle around (0,0) with radius A088959(n).

Original entry on oeis.org

4, 12, 20, 36, 60, 108, 180, 252, 324, 540, 756, 972, 1620, 2268, 2916, 4860, 6804, 8748, 14580, 20412, 26244, 43740, 61236, 72900, 78732, 102060, 131220, 183708, 218700, 236196, 306180, 393660, 551124, 656100, 708588, 918540

Offset: 1

Günter Rote, Sep 12 2023

nonn

Sequence of records of A046109 (first term 1 from A046109 is omitted).

See A071385 for radii that are not necessarily integers.

a(n) = 8*A088111(n) + 4.

cp's OEIS Frontend

A088899 T(n, k) = number of ordered pairs of integers (x,y) with x^2/n^2 + y^2/k^2 = 1, 1 <= k <= n; triangular array, read by rows.

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A292276 a(n) = number of vertices of the convex hull of the set of points of norm <= n^2 in square lattice.

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A046110 Number of lattice points on circumference of a circle of radius (2n+1)/2 with center at (1/2,0).

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A046111 Number of lattice points on circumference of a circle of radius 1/3,2/3,4/3,5/3,... with center at (1/3,0).

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A256452 Number of integer solutions to n^2 = x^2 + y^2 with x>0, y>=0.

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Maple

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A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

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C

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A355760 a(n) is the number of grid points in a square lattice covered by the area enclosed by n loops of an Archimedean spiral with starting point (0, 0) and endpoint (n, 0).

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A350459 Number of positive rational points on the unit circle with denominator <= n and numerator >= 1.

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Python

A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.

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