A001481 -id:A001481 - cp's OEIS frontend

A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

0, 1, 4, 9, 13, 16, 25, 36, 37, 49, 52, 61, 64, 73, 81, 97, 100, 109, 117, 121, 144, 148, 157, 169, 181, 193, 196, 208, 225, 229, 241, 244, 256, 277, 289, 292, 313, 324, 325, 333, 337, 349, 361, 373, 388, 397, 400, 409, 421, 433, 436, 441, 457, 468, 481, 484

Offset: 1

M. F. Hasler, Jan 24 2009

Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Ron Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys. 69, 1181 (1997). This sequence coincides with the row N = 12 of Table VII.

Cf. A000290, A001481, A003136, A155561.

isA155563(n,/* use optional 2nd arg to get other analogous sequences */c=[3,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155563(n) & print1(n","))

is(n)=(n==0) || (#bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n));
select(n->is(n), vector(1500,j,j-1)) \\ Joerg Arndt, Jan 11 2015

from itertools import count, islice
from sympy import factorint
def A155563_gen(): # generator of terms
    return filter(lambda n: all(e & 1 == 0 or (p & 3 != 3 and p % 3 < 2) for p, e in factorint(n).items()),count(0))
A155563_list = list(islice(A155563_gen(),30)) # Chai Wah Wu, Jun 27 2022

A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 5, 9, 16, 20, 25, 29, 36, 41, 45, 49, 61, 64, 80, 81, 89, 100, 101, 109, 116, 121, 125, 144, 145, 149, 164, 169, 180, 181, 196, 205, 225, 229, 241, 244, 245, 256, 261, 269, 281, 289, 305, 320, 324, 349, 356, 361, 369, 389, 400, 401, 404, 405, 409, 421

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155575 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155565(n,/* use optional 2nd arg to get other analogous sequences */c=[5,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155565(n) & print1(n","))

A155566 Intersection of A001481 and A002481: N = a^2 + b^2 = c^2 + 6d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 25, 36, 40, 49, 58, 64, 73, 81, 90, 97, 100, 106, 121, 144, 145, 160, 169, 193, 196, 202, 225, 232, 241, 250, 256, 265, 289, 292, 298, 313, 324, 337, 346, 360, 361, 388, 394, 400, 409, 424, 433, 441, 457, 484, 490, 505, 522, 529, 538, 576

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155576 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155566(n,/* use optional 2nd arg to get other analogous sequences */c=[6,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,600, isA155566(n) & print1(n","))

A155568 Intersection of A001481 inter A020670: N = a^2 + b^2 = c^2 + 7d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 25, 29, 32, 36, 37, 49, 53, 64, 72, 81, 100, 109, 113, 116, 121, 128, 137, 144, 148, 149, 169, 193, 196, 197, 200, 212, 225, 232, 233, 256, 261, 277, 281, 288, 289, 296, 317, 324, 333, 337, 361, 373, 389, 392, 400, 401, 421, 424, 436, 441, 449

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155578 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155568(n,/* use optional 2nd arg to get other analogous sequences */c=[7,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155568(n) & print1(n","))

A173256 Partial sums of A001481.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 29, 39, 52, 68, 85, 103, 123, 148, 174, 203, 235, 269, 305, 342, 382, 423, 468, 517, 567, 619, 672, 730, 791, 855, 920, 988, 1060, 1133, 1207, 1287, 1368, 1450, 1535, 1624, 1714, 1811, 1909, 2009, 2110, 2214, 2320, 2429, 2542, 2658, 2775

Offset: 1

Jonathan Vos Post, Feb 14 2010

nonn

The subsequence of primes in this sequence begins 3, 7, 29, 103, 269, 619, 1811, 3271.

			a(66) = 0 + 1 + 2 + 4 + 5 + 8 + 9 + 10 + 13 + 16 + 17 + 18 + 20 + 25 + 26 + 29 + 32 + 34 + 36 + 37 + 40 + 41 + 45 + 49 + 50 + 52 + 53 + 58 + 61 + 64 + 65 + 68 + 72 + 73 + 74 + 80 + 81 + 82 + 85 + 89 + 90 + 97 + 98 + 100 + 101 + 104 + 106 + 109 + 113 + 116 + 117 + 121 + 122 + 125 + 128 + 130 + 136 + 137 + 144 + 145 + 146 + 148 + 149 + 153 + 157 + 160 = 4876.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544, A004018, A000161, A002654, A064533, A000404, A002828, A000378, A025284-A025320, A125110, A091072.

N:= 1000:
A001481:= sort(convert({seq(seq(x^2+y^2, y=0..floor(sqrt(N-x^2))),x=0..floor(sqrt(N)))},list)):
ListTools:-PartialSums(A001481); # Robert Israel, Mar 15 2016

from itertools import count, accumulate, islice
from sympy import factorint
def A173256_gen(): # generator of terms
    return accumulate(filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0)))
A173256_list = list(islice(A173256_gen(),30)) # Chai Wah Wu, Jun 27 2022

a(n) = Sum_{i=1..n} A001481(i) = Sum_{i=1..n} (numbers that are the sum of 2 nonnegative squares) = Sum_{i=1..n} (numbers n such that i = x^2 + y^2 has a solution in nonnegative integers x, y).

a(21) corrected by Robert Israel, Mar 15 2016

A272268 Records in A001481 that are more than twice the previous record.

Original entry on oeis.org

1, 4, 9, 20, 41, 85, 173, 349, 701, 1405, 2813, 5629, 11261, 22525, 45053, 90113, 180232, 360466, 720937, 1441877, 2883761, 5767525, 11535053, 23070112, 46140228, 92280457, 184560921, 369121849, 738243700, 1476487402, 2952974809, 5905949620, 11811899241

Offset: 1

M. Sinan Kul, Apr 24 2016

nonn

This list can be seen as the r^2 (square of the radius r) of the homocentric circles that are centered at the origin and pass through at least 4 lattice points, the innermost circle being the unit circle.
If we start with the unit circle (x^2 + y^2 = 1), the smallest circle that pass through at least four lattice points would be x^2 + y^2 = 4 with (2,0), (0,2), (-2,0), (0,-2). Similarly next circle would be x^2 + y^2 = 9 passing through (3,0), (0,3), (-3,0), (0,-3), and the next x^2 + y^2 = 20 passing through (2,4), (4,2), (-2,4), (-4,2), (-2,-4), (-4,-2), (2,-4), (4,-2), etc.
It is also worth mentioning that a square can be drawn with vertexes on the lattice points of a circle and the sides of that square wouldn't touch the smaller circle.

M. Sinan Kul, PARI/GP code

Cf. A001481.

NestList[SelectFirst[Range[2 # + 1, 5 #], SquaresR[2, #] > 0 &] &, 1, 25] (* Michael De Vlieger, Apr 25 2016, Version 10 *)

A296147 Number of shapes of grid-filling curves of order A001481(n) (on the square grid) with turns by +-90 degrees that are generated by folding morphisms.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 20, 14, 44, 32, 69, 212, 287, 796, 438, 1402, 4232, 3202, 2242, 14316, 5080, 11122, 12374, 155305, 152602, 77469

Offset: 1

Joerg Arndt and Julia Handl, Dec 06 2017

a(1) and a(2) correspond to the trivial (empty and single-stroke) curves of orders 0 and 1 respectively.

Jörg Arndt, Julia Handl, Plane-filling Folding Curves on the Square Grid, Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture; Stockholm (2018).

Cf. A296148 (same sequence, including zero terms).

Cf. A265685 (simple curves of order 4*n+1).

A328801 Least k such that there exists a square of side length sqrt(A001481(n)) with vertices in a k X k square array of points.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 6, 5, 6, 7, 7, 6, 7, 8, 9, 9, 7, 8, 9, 10, 10, 8, 9, 11, 10, 11, 12, 9, 10, 11, 13, 12, 13, 13, 10, 11, 12, 14, 13, 14, 15, 11, 12, 13, 15, 14, 16, 15, 16, 12, 13, 14, 17, 15, 17, 16, 13, 14, 17, 15, 18, 16, 18, 17, 19, 19, 14, 15, 16, 17

Offset: 2

Peter Kagey, Oct 27 2019

			For n = 8, there is a square with side length sqrt(A001481(8)) = sqrt(10) and vertices in the a(8) X a(8) = 5 X 5 square array of points.
o o o * o
* o o o o
o o o o o
o o o o *
o * o o o
However, there is no square with side length sqrt(10) and vertices in a smaller square array points.

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A001481, A108279.

A328793 is the analog for a triangular grid.

a(n) = A328803(n) + 1.

A328804 The maximum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 4, 5, 6, 6, 7, 6, 7, 8, 8, 6, 7, 8, 9, 9, 7, 10, 10, 9, 10, 11, 8, 11, 10, 12, 11, 12, 12, 9, 10, 13, 13, 12, 13, 14, 14, 11, 12, 14, 13, 15, 14, 15, 11, 12, 15, 16, 16, 16, 15, 12, 17, 16, 14, 17, 15, 17, 16, 18, 18, 17, 18, 15, 16

Offset: 1

Peter Kagey, Oct 27 2019

nonn

			For n = 14, A001481(14) = 25 = 0^2 + 5^2 = 3^2 + 4^2, so a(14) = max{0+5, 3+4} = 7.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000161, A001481, A328803.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
from sympy import factorint
def A328804_gen(): # generator of terms
    return map(lambda n: max((a+b for a, b in diop_DN(-1,n))), filter(lambda n:(lambda m:all(d&3!=3 or m[d]&1==0 for d in m))(factorint(n)), count(0)))
A328804_list = list(islice(A328804_gen(),30)) # Chai Wah Wu, Sep 09 2022

A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

Original entry on oeis.org

0, 4, 8, 12, 12, 16, 20, 20, 20, 28, 28, 32, 28, 28, 36, 36, 40, 36, 44, 44, 44, 44, 44, 52, 48, 52, 52, 52, 52, 60, 52, 60, 64, 60, 60, 60, 68, 68, 60, 68, 68, 68, 72, 68, 76, 76, 76, 76, 76, 76, 76, 84, 84, 76, 88, 76, 84, 84, 92, 84, 92

Offset: 1

Scott R. Shannon, Mar 28 2020

nonn

Draw a circle on a 2D square grid centered at the origin with a radius squared equal to the norm of the Gaussian integers A001481(n). See the images in the links. This sequence gives the number of unit cells intersected by the circumference of the circle. Equivalently this is the number of intersections of the circumference with the x and y integer grid lines.

Scott R. Shannon, Illustration for n = 3. The circle has a radius squared of 2, resulting in 8 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 4. The circle has a radius squared of 4, resulting in 12 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 8. The circle has a radius squared of 10, resulting in 20 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 12. The circle has a radius squared of 18, resulting in 32 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 13. The circle has a radius squared of 20, resulting in 28 unit cells intersected/intersection points. This is the first term where the number of intersection points decreases relative to the previous term.
Scott R. Shannon, Illustration for n = 26. The circle has a radius squared of 52, resulting in 52 unit cells intersected/intersection points.
Wikipedia, Gaussian integer.

Cf. A001481, A055025, A057655, A119439, A242118 (a subsequence of this sequence), A234300.

a(n) = 4*A234300(2*(n-1)). - Andrey Zabolotskiy, Feb 22 2025

cp's OEIS Frontend

A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

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A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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PARI

A155566 Intersection of A001481 and A002481: N = a^2 + b^2 = c^2 + 6d^2 for some integers a,b,c,d.

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PARI

A155568 Intersection of A001481 inter A020670: N = a^2 + b^2 = c^2 + 7d^2 for some integers a,b,c,d.

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PARI

A173256 Partial sums of A001481.

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Maple

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Extensions

A272268 Records in A001481 that are more than twice the previous record.

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Mathematica

A296147 Number of shapes of grid-filling curves of order A001481(n) (on the square grid) with turns by +-90 degrees that are generated by folding morphisms.

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A328801 Least k such that there exists a square of side length sqrt(A001481(n)) with vertices in a k X k square array of points.

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A328804 The maximum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

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