A057653 -id:A057653 - cp's OEIS frontend

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

2, 10, 18, 26, 34, 50, 58, 74, 82, 90, 98, 106, 122, 130, 146, 162, 170, 178, 194, 202, 218, 226, 234, 242, 250, 274, 290, 298, 306, 314, 338, 346, 362, 370, 386, 394, 410, 442, 450, 458, 466, 482, 490, 514, 522, 530, 538, 554, 562, 578, 586, 610, 626, 634

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and complement of A024352 (difference of squares).
Numbers of the form 4k+2 = double of an odd number, with the odd number equal to the sum of 2 squares (sequence A057653). - Jean-Christophe Hervé, Oct 24 2015
Numbers that are the sum of two odd squares. - Jean-Christophe Hervé, Oct 25 2015

			2 = 1^2 + 1^2, 10 = 1^2 + 3^2, 18 = 3^2 + 3^2.

Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097268, A097270, A097271. Equals twice A057653.

Cf. A020668, A263715.

is(n)=if(n%4!=2,return(0)); my(f=factor(n/2)); for(i=1,#f[,1],if(bitand(f[i,2],1)==1&&bitand(f[i,1],3)==3, return(0))); 1 \\ Charles R Greathouse IV, May 31 2013

from itertools import count, islice
from sympy import factorint
def A097269_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n//2).items()),count(2,4))
A097269_list = list(islice(A097269_gen(),30)) # Chai Wah Wu, Jun 28 2022

A234334 Numbers k such that both distances from k to two nearest squares are perfect squares.

Original entry on oeis.org

0, 1, 5, 8, 25, 40, 45, 65, 80, 153, 160, 169, 200, 221, 325, 360, 416, 425, 493, 520, 625, 680, 725, 925, 936, 1025, 1040, 1073, 1088, 1305, 1360, 1681, 1768, 1800, 1813, 1845, 1961, 2000, 2320, 2385, 2501, 2600, 2925, 3016, 3185, 3200, 3400, 3445, 3721, 3848

Offset: 1

Alex Ratushnyak, Dec 23 2013

Except a(1)=0, a(n) are numbers k such that both k-x and y-k are perfect squares, where x and y are two nearest to k squares: x < k <= y.
The sequence of sums of distances begins: 1, 1, 5, 5, 9, 13, 13, 17, 17, 25, 25, 25, 29, 29, 37, 37, 41, 41, 45, 45, 49, 53, 53, 61, 61, 65, 65, 65, 65, 73, 73, 81, 85, ... (cf. A057653).
Each term is either a square or has a pair: if i^2 + j^2 = 2*m+1 then m^2+i^2 and m^2+j^2 are both in the sequence.

			The two squares nearest to 25 are 16 and 25, because both 25-25=0 and 25-16=9 are squares, 25 is in the sequence.
The two squares nearest to 45 are 36 and 49, because both 45-36=9 and 49-45=4 are squares, 45 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A000290.
Cf. subsequences: A007204, A234335, A234336.
Cf. A234348.

filter:= proc(n) local a;
  if issqr(n) then a:= sqrt(n)-1 else a:= floor(sqrt(n)) fi;
  issqr(n-a^2) and issqr((a+1)^2-n)
end proc:
select(filter, [$0..5000]); # Robert Israel, Jan 21 2021

filter[n_] := If[n == 0, True, Module[{a}, a = If[IntegerQ @ Sqrt[n], Sqrt[n]-1, Floor[Sqrt[n]]]; IntegerQ @ Sqrt[n-a^2] && IntegerQ@Sqrt[(a+1)^2-n]]];
Select[Range[0, 5000], filter] (* Jean-François Alcover, May 28 2023, after Robert Israel *)

A263715 Nonnegative integers that are the sum or difference of two squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80

Offset: 1

Jean-Christophe Hervé, Oct 24 2015

nonn

Contains all integers that are not equal to 2 (mod 4) (they are of the form y^2 - x^2) and those of the form 4k+2 = 2*(2k+1) with the odd number 2k+1 equal to the sum of two squares (A057653).

			2 = 1^2 + 1^2, 3 = 2^2 - 1^2, 4 = 2^2 + 0^2, 5 = 2^2 + 1^2 = 3^2 - 2^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A001481, A057653, A097269, A042965, A020668, A263737.

Cf. A062316 (complement), A079298.

r[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers] || Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[r[n] =!= False, Sow[n]], {n, 0, 80}]][[2, 1]] (* Jean-François Alcover, Oct 25 2015 *)

from itertools import count, islice
from sympy import factorint
def A263715_gen(): # generator of terms
    return filter(lambda n: n & 3 != 2 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A263715_list = list(islice(A263715_gen(),30)) # Chai Wah Wu, Jun 28 2022

Union of A001481 (sums of two squares) and A042965 (differences of two squares).
Union of A042965 and 2*A057653 = A097269, with intersection of A042965 and A097269 = {}.
Union of A020668 (x^2+y^2 and a^2-b^2), A097269 (x^2+y^2, not a^2-b^2) and A263737 (not x^2+y^2, a^2-b^2).

A234434 Number of shapes of grid-filling curves (on the triangular grid) with turns by 0, +120, or -120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

Original entry on oeis.org

1, 1, 0, 0, 3, 0, 5, 0, 0, 10, 15, 0, 0, 17, 0, 0, 71, 0, 213, 0, 0, 0, 184, 0, 549, 845, 0, 0, 1850, 0, 0, 0, 0, 6700, 9787, 0, 30475, 0, 0, 0, 52184, 0, 0, 0, 0, 182043, 401377, 0, 0, 604809, 0, 0, 0, 0, 4318067, 0, 0, 0, 7158120, 0

Offset: 3

Joerg Arndt, Dec 26 2013

Shapes are considered modulo reflections and rotations.
The curves considered are not self-intersecting, not edge-contacting (i.e., have double edges), but (necessarily) vertex-contacting (i.e., a point in the grid is visited more than once).
The L-systems are interpreted as follows: 'F' is a unit-stroke in the current direction, '+' is a turn left by 120 degrees, '-' a turn right by 120 degrees, and '0' means "no turn".
The images in the links section use rounded corners to make the curves visually better apparent.
Three copies of each curve (connected by three turns '+' or three turns '-') give two tiles (that tile the triangular grid), but symmetric curves (any symmetry) give just one tile(-shape). The tiles are 3-symmetric, and sometimes (only for n of the form 6*k+1) 6-symmetric. There could in general be more tile-shapes than curve-shapes, for n=7 both cardinalities coincide, see links section. It turns out that for large n there are actually fewer tile-shapes than curve-shapes.
Terms a(n) are nonzero for n>=3 if and only if n is a term of A003136.
The equivalent sequence for the square grid has nonzero terms for n>=5 that are terms of A057653.
If more symbols are allowed for the L-systems, more curves are found, also if strokes of lengths other than one unit are allowed, see the Ventrella reference.
For n = 49 there are two pairs (x, y) such that x^2 + x*y + y^2 = n, (7, 0) and (5, 3), respectively giving 132271 and 269106 shapes (a(49) = 401377 = 132271 + 269106). The next n with two such pairs (x, y) is n = 91, with pairs (6, 5) and (9, 1) - Joerg Arndt, Apr 07 2019

			The a(3)=1 shape of order 3 is generated by F |--> F+F-F, the curve generated by F |--> F-F+F has the same shape (after reflection). The curve is called the "terdragon", see A080846.
There are 5 L-systems that generate a curve of order 7 with first turn '0' or '+':
F |--> F0F+F0F-F-F+F  #  R7-1
F |--> F0F+F+F-F-F0F  #  R7-2
F |--> F+F0F+F-F0F-F  #  R7-3
F |--> F+F-F-F0F+F0F  #  R7-4 # same shape as R7-1
F |--> F+F-F-F+F+F-F  #  R7-5 # same shape as R7-2
As shown, these give just 3 shapes (and the L-systems with first turn '-' give no new shapes), so a(7)=3.
The curve R7-1 appears on page 107 in the Ventrella reference.
The symmetric curves R7-2 and R7-5 appear in the Arndt reference (there named "R7-dragon" and "second R7-dragon", see A176405 and A176416).

Joerg Arndt, Matters Computational (The Fxtbook), see section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, images of the R7-dragons are given on p. 97 and p. 98
Joerg Arndt, all 3 shapes of curves of order 7, rendered after 4 generations of the L-systems.
Joerg Arndt, all 3 shapes of tiles of order 7, rendered after 4 generations of the L-systems, curves colored to make them apparent.
Joerg Arndt, all 15 shapes of curves of order 13, rendered after 3 generations of the L-systems (file size about 500 kB).
Joerg Arndt, all shapes of tiles of order 13, rendered after 3 generations of the L-systems (file size about 500 kB). Note: not all symmetries are accounted for, so some tiles appear more than once (e.g., in flipped over form).
Joerg Arndt, decompositions of order-13 curves into self-similar parts (file size about 1.3 MB)
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Jeffrey J. Ventrella, Brain-Filling Curves: A Fractal Bestiary, 2012.

Cf. A265685 (shapes on the square grid), A265686 (tri-hexagonal grid).

Terms a(21), a(27), a(28), and a(31) corrected by Joerg Arndt, Jun 20 2018
Terms a(32) - a(47) from Joerg Arndt, Jun 22 2018
Terms a(48) - a(51)  from Joerg Arndt, Nov 18 2018
Terms a(52) - a(56) added and a(48) - a(49) corrected, Joerg Arndt, Apr 07 2019
Terms a(57) - a(62) from Joerg Arndt, Apr 10 2019

A265685 Number of shapes of grid-filling curves of order 4*n+1 (on the square grid) with turns by +-90 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

Original entry on oeis.org

1, 1, 4, 6, 0, 33, 39, 0, 164, 335, 0, 603, 2467, 0, 10412, 19692, 0, 79494, 0, 155155, 1271455, 1272243, 0

Offset: 1

Joerg Arndt, Dec 13 2015

Such curves exist only for n such that 4*n+1 is a term of A057653.

Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Jörg Arndt, Julia Handl, Plane-filling Folding Curves on the Square Grid, 2018. (This paper is about a different class of curves where the Lindenmayer-systems have two no-constant symbols)

Cf. A234434 (shapes on the triangular grid), A265686 (tri-hexagonal grid).
Cf. A296148 (folding curves of order n) and A296149 (folding curves of order 4*n+1).
Cf. A306358 (curve orders with at least two decomposition x^2 + y^2).

a(15)..a(23) from Joerg Arndt, Feb 12 2019

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A339952 Numbers that are the sum of an even square > 0 and an odd square.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 45, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 117, 125, 137, 145, 149, 153, 157, 169, 173, 181, 185, 193, 197, 205, 221, 225, 229, 233, 241, 245, 257, 261, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 333, 337, 349, 353, 365, 369, 373, 377

Offset: 1

Wesley Ivan Hurt, Dec 24 2020

			13 is in the sequence since it is the sum of an even square > 0 and an odd square, 2^2 + 3^2 = 4 + 9 = 13.

Cf. A000404, A010052, A057653, A097269.

Table[If[Sum[Sign[(Mod[i, 2] Mod[n - i + 1, 2] + Mod[i + 1, 2] Mod[n - i, 2])] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 500}] // Flatten

def aupto(limit):
  m = int(limit**.5) + 2
  es = [i*i for i in range(2, m, 2)]
  os = [i*i for i in range(1, m, 2)]
  return sorted(set(a+b for a in es for b in os if a+b <= limit))
print(aupto(377)) # Michael S. Branicky, May 13 2021

A280965 Nonsquares whose distances to the two nearest squares are squares.

Original entry on oeis.org

5, 8, 40, 45, 65, 80, 153, 160, 200, 221, 325, 360, 416, 425, 493, 520, 680, 725, 925, 936, 1025, 1040, 1073, 1088, 1305, 1360, 1768, 1800, 1813, 1845, 1961, 2000, 2320, 2385, 2501, 2600, 2925, 3016, 3185, 3200, 3400, 3445, 3848, 3869, 3944, 3965, 4640, 4745, 5185, 5248, 5265, 5328, 5525, 5576, 5785, 5920, 6120

Offset: 1

Emmanuel Vantieghem, Feb 27 2017

nonn

The sequence is infinite because there are terms of it between n^2 and (n+1)^2 whenever 2n+1 is a sum of two squares.

			a(3) = 40 because the two nearest squares are 36 and 49 and 40 - 36 = 4, 49 - 40 = 9 are both squares.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A057653, A234334.

Select[Range[6120], IntegerQ[Sqrt[# - (Floor[Sqrt[#]])^2]] && IntegerQ[Sqrt[(Ceiling[Sqrt[#]])^2 - #]] &]

is(n)=my(k=sqrtint(n)); issquare(n-k^2) && issquare((k+1)^2-n) && n>k^2 \\ Charles R Greathouse IV, Feb 27 2017

list(lim)=my(v=List(),k2,K2,n); for(k=2,sqrtint(lim\1)-1, k2=k^2; K2=(k+1)^2; for(s=1,sqrtint(K2-k2-1), n=k2+s^2; if(issquare(K2-n), listput(v,n)))); k2=sqrtint(lim\1)^2; K2=(sqrtint(lim\1)+1)^2; for(n=k2+1,lim, if(issquare(n-k2) && issquare(K2-n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Feb 27 2017

A296149 Number of shapes of grid-filling curves of order 4*n+1 (on the square grid) with turns by +-90 degrees that are generated by folding morphisms.

Original entry on oeis.org

2, 3, 14, 32, 0, 287, 438, 0, 2242, 5080, 11122, 12374, 77469

Offset: 1

Joerg Arndt, Dec 06 2017

Terms are nonzero if and only if 4*n+1 is a term of A057653.

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 (number of folding curves of all orders).

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

A339273 Sums of two nonzero even squares.

Original entry on oeis.org

8, 20, 32, 40, 52, 68, 72, 80, 100, 104, 116, 128, 136, 148, 160, 164, 180, 200, 208, 212, 232, 244, 260, 272, 288, 292, 296, 320, 328, 340, 356, 360, 388, 392, 400, 404, 416, 424, 436, 452, 464, 468, 488, 500, 512, 520, 544, 548, 580, 584, 592, 596, 612, 628, 640, 648, 656

Offset: 1

Wesley Ivan Hurt, Dec 24 2020

			20 is in the sequence since it is the sum of two nonzero even squares, 2^2 + 4^2 = 4 + 16 = 20.

Cf. A000404, A010052, A057653, A097269.

Table[If[Sum[Mod[i + 1, 2] Mod[n - i + 1, 2] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 700}] // Flatten

a(n) = 4*A000404(n).

Characteristic function: sign(Sum_{k=1..floor(n/2)} ((k+1) mod 2) * ((n-k+1) mod 2) * c(k) * c(n-k)), where c is the characteristic function of squares (A010052).

cp's OEIS Frontend

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

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A234334 Numbers k such that both distances from k to two nearest squares are perfect squares.

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A263715 Nonnegative integers that are the sum or difference of two squares.

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A234434 Number of shapes of grid-filling curves (on the triangular grid) with turns by 0, +120, or -120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

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A265685 Number of shapes of grid-filling curves of order 4*n+1 (on the square grid) with turns by +-90 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

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A294774 a(n) = 2*n^2 + 2*n + 5.

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A339952 Numbers that are the sum of an even square > 0 and an odd square.

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A280965 Nonsquares whose distances to the two nearest squares are squares.

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A294774 a(n) = 2n^2 + 2n + 5.