A157228 -id:A157228 - cp's OEIS frontend

A025441 Number of partitions of n into 2 distinct nonzero squares.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0

Offset: 0

David W. Wilson

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences related to sums of squares

Cf. A060306 gives records; A052199 gives where records occur.
Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.
Column k=2 of A341040.
Cf. A004439 (a(n)=0), A025302 (a(n)=1), A025303 (a(n)=2), A025304 (a(n)=3), A025305 (a(n)=4), A025306 (a(n)=5), A025307 (a(n)=6), A025308 (a(n)=7), A025309 (a(n)=8), A025310 (a(n)=9), A025311 (a(n)=10), A004431 (a(n)>0).

a025441 n = sum $ map (a010052 . (n -)) $
                      takeWhile (< n `div` 2) $ tail a000290_list
-- Reinhard Zumkeller, Dec 20 2013

Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* Jean-François Alcover, Mar 01 2019 *)

a(n)=if(n>4,sum(k=1,sqrtint((n-1)\2),issquare(n-k^2)),0) \\ Charles R Greathouse IV, Jun 10 2016

a(n)=if(n<5,return(0)); my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2-issquare(n/2) \\ Charles R Greathouse IV, Jun 10 2016

from math import prod
from sympy import factorint
def A025441(n):
    f = factorint(n).items()
    return -int(not (any((e-1 if p == 2 else e)&1 for p,e in f) or n&1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

a(A025302(n)) = 1. - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019
a(n) = Sum_{i=1..floor((n-1)/2)} c(i) * c(n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020
a(n) = A000161(n) - A093709(n). - Andrey Zabolotskiy, Apr 12 2022

A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

Original entry on oeis.org

1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 10, 13, 12, 18, 9, 22, 9, 21, 14, 16, 14, 29, 11, 17, 16, 29, 12, 28, 12, 25, 23, 20, 13, 39, 16, 27, 20, 29, 15, 34, 20, 36, 22, 25, 16, 50, 17, 26, 29, 38, 24, 40, 18, 36, 26, 40

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.
The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145394 (p6), A003051 (p6mm).
Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see table 6 and fig. 2).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2; beware the typo in a(5).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A054345, A054346, A167156, A008621.
Cf. A000203, A069734, A145391, A145392, A145394, A003051, A002324, A002654, A069735, A145390.
Cf. A019590, A157228, A157226, A157230, A157231, A053866, A025441.

terms = 70;
CoefficientList[Sum[(1/((1-x^m)(1-x^(4m)))-1), {m, 1, terms}] + O[x]^(terms + 1), x] // Rest (* Jean-François Alcover, Aug 05 2018 *)

a(n) = (A000203(n) + A002654(n) + A069735(n) + A145390(n))/4. [Rutherford] - N. J. A. Sloane, Mar 13 2009
G.f.: Sum_{ m>=1 } (1/((1-x^m)(1-x^(4m))) - 1). [Hanany, Orlando & Reffert, eq. (6.8)] - Andrey Zabolotskiy, Jul 05 2017
a(n) = Sum_{ m: m^2|n } A019590(n/m^2) + A157228(n/m^2) + A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2) = A053866(n) + A025441(n) + Sum_{ m: m^2|n } A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2). [Rutherford] - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008621(d) = Sum_{ d|n } (1 + floor(d/4)). [From the above-given g.f.] - Andrey Zabolotskiy, Jul 17 2019

New name from Andrey Zabolotskiy, Mar 12 2018

A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

Original entry on oeis.org

1, 3, 2, 4, 2, 6, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 2, 6, 2, 8, 4, 6, 2, 8, 2, 6, 2, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 8, 4, 6, 2, 8, 2, 6, 4, 8, 2, 6, 4, 8, 4, 6, 2, 16, 2, 6, 4, 4, 4, 12, 2, 8, 4, 12, 2, 8, 2, 6, 4, 8, 4, 12, 2, 8, 2, 6, 2, 16, 4

Offset: 1

Andrey Zabolotskiy, May 07 2018

			There are 6 = A001615(4) lattices in Z^2 whose quotient group is C_4. The reflection through an axis relates <(4,0), (1,1)> and <(4,0), (3,1)>. The remaining 4 = a(4) lattices are fixed.

Álvar Ibeas, Table of n, a(n) for n = 1..10000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4. Contains errors for n = 24 and 28.]

Cf. A069735 (not only primitive sublattices), A304183 (primitive oblique sublattices), A069734 (all sublattices).
Cf. other columns of tables 4 and 5 from [Rutherford, 2009]: A001615, A060594, A157223, A000089, A157224, A000086, A157227, A019590, A157228, A157226, A157230, A157231, A154272, A157235.
Cf. A034444, A001221, A001620, A306016.

f[p_, e_] := If[p == 2, If[e == 1, 3, 4], 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

From Álvar Ibeas, Mar 18 2021: (Start)
For n odd, a(n) = A034444(n) = 2^(A001221(n)).
For n even, a(n) = A034444(n) + A034444(n/2). If 4|n, a(n) = 2^(A001221(n) + 1); otherwise, a(n) = 3 * 2^(A001221(n) - 1).
Multiplicative with a(2) = 3, a(2^e) = 4 (for e>1), and a(p^e) = 2 (for p>2).
Dirichlet g.f.: (1+2^(-s)) * zeta(s)^2 / zeta(2s).
(End)
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*9*n/Pi^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 31 2022

A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1, 4, 2, 1, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

After a(2), this matches A034380 except for n = 63, 65, 80, 85, ... - R. J. Mathar, Feb 27 2009 [Updated by Andrey Zabolotskiy, May 09 2018]

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Act. Cryst. A65 (2009) 156-163, Table 5 symmetry *mm2.

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157231, A304182, A060594, A046072, A033948, A272592.

a[n_] := If[n <= 2, 0, Sum[Boole[Mod[k^2, n] == 1], {k, 1, n}]/2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 12 2023 *)

From Andrey Zabolotskiy, Sep 30 2018: (Start)
a(n) = (A060594(n) - A019590(n))/2.
a(n) = 2^(A046072(n)-1) for n>2. Thus a(n) = 1 if n>2 is in A033948, a(n) = 2 if n is in A272592, etc. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 4, 2, 3, 1, 4, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 1, 8, 1, 3, 2, 2, 2, 6, 1, 4, 2, 6, 1, 4, 1, 3, 2, 4, 2, 6, 1, 4, 1, 3, 1, 8, 2, 3, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy's new formula confirms that a(n) indeed is a function of A305891(n). - Antti Karttunen, Oct 01 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.

A007875(n) = eulerphi(2^omega(n));
A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ Antti Karttunen, Oct 01 2018

From Andrey Zabolotskiy, Sep 30 2018: (Start)
Let b(n) = A007875(n) for n>1, b(1) = 0. Then
a(n) = b(n) for odd n,
a(n) = b(n) + b(n/2) for even n.
Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 3, 7, 4, 6, 6, 7, 5, 8, 6, 8, 8, 9, 6, 14, 7, 10, 10, 11, 10, 15, 8, 13, 12, 14, 9, 20, 10, 15, 16, 16, 11, 20, 13, 20, 16, 18, 12, 25, 16, 20, 18, 20, 14, 30, 14, 22, 22, 22, 18, 32, 16, 24, 22, 32, 17, 32, 17

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157230, A001615, A304182.

a(n) = (A001615(n) - A019590(n) - 2 * (A157228(n) + A157226(n) + A157230(n))) / 4. - Andrey Zabolotskiy, May 09 2018

New name and more terms from Andrey Zabolotskiy, May 09 2018

A193138 Number of square satins of order n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 3

N. J. A. Sloane, Jul 16 2011

nonn

a(n) = A157228(n) for all entries known. - R. J. Mathar, Aug 10 2011
This sequence is conjectured to coincide with the multiplicities of the representation of n >= 3 as primitive sums of two squares. Neither the order of the squares nor the signs of the numbers to be squared are taken into account. a(n) = 0  if no such representation exists. Checked for n = 3,4, ..., 1000 (using the program below). The two squares are in each case nonzero and distinct. If one includes also 0 as a square in the primitive sum of two squares one could take a(0) = 0, a(1) = 1, a(2) = 1. If only nonzero squares are considered, then one could take a(0) = 0, a(1) = 0, a(2) = 1.
For the numbers n with a(n) > 0 (in this conjectured interpretation of a(n)) see A008784. - Wolfdieter Lang, Apr 17 2013
The stated conjecture is true because it follows immediately from Theorem 3.22, p. 165, of the Niven-Zuckerman-Montgomery reference. There r(n) gives the number of primitive solutions of n = x^2 + y^2 with ordered and signed pairs of integers x,y. Because x and y are distinct if n >= 3 one needs here a(n) = r(n)/2^3. This then coincides with the formula for u(n) given in the Grünbaum-Shephard Theorem 5. - Wolfdieter Lang, Apr 18 2013
The equality noted by R. J. Mathar above indeed holds for all n > 2. Regarding n = 2 case: if we consider periodic twills as satins (which seems more consistent), we'll get a(2) = 1 from the plain weave; otherwise (following Grünbaum and Shephard), a(1) = a(2) = 0 (so we get A157228). In the former case, the all-black pattern can formally be counted as a(1) = 1, but physically it is dubious (this pattern corresponds to unweaved warp and weft). - Andrey Zabolotskiy, May 09 2018

			Primitive sums of two squares stated as a comment above: a(3) = 0  because 3 is not a sum of two squares.  a(5) = 1 because 5 = 1^2 + 2^2, denoted by the unique (primitive) doublet [1, 2].  a(65) = 2 from the two (primitive) doublets [1, 8] and [4, 7]. a(85) = 2 with the (primitive) doublets [2, 9] and  [6, 7]. a(8) = 0 because the doublet [2, 2] is imprimitive. - _Wolfdieter Lang_, Apr 18 2013

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

B. Grünbaum and G. C. Shephard, Satins and twills: an introduction to the geometry of fabrics, Math. Mag., 53 (1980), 139-161. See Theorem 5, p. 152.

Cf. A193139, A193140, A157228.

U:=proc(n) local nop,p3,i,t1,t2,al,even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
nop:=t2-even;
p3:=0;
for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:
if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(nop-1)); fi;
end;
[seq(U(n),n=3..120)];

a[n_] := Select[ PowersRepresentations[n, 2, 2], GCD @@ # == 1 &] // Length; a[2] = 0; Table[a[n], {n, 3, 120}] (* Jean-François Alcover, Apr 18 2013 *)

Take the prime number factorization (symbolically) as n = 2^a*product(p^b)*product(q^c) with primes p == 1(mod 4) and primes q == 3(mod 4) and n>=3. If a = 0 or 1 and all c's vanish then a(n) = 2^(t-1) with t the number of distinct primes congruent 1(mod 4). Otherwise a(n) = 0. (See the Niven-Zuckerman-Montgomery reference, Theorem 3.22, p. 165, and the Grünbaum-Shephard Theorem 5 formula for u(n)). - Wolfdieter Lang, Apr 18 2013

cp's OEIS Frontend

A025441 Number of partitions of n into 2 distinct nonzero squares.

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A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

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A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

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A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

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A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

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A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

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A193138 Number of square satins of order n.

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