A193138 -id:A193138 - cp's OEIS frontend

A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

0, 0, 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

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

From Andrey Zabolotskiy, May 09 2018: (Start)
Also, the number of partitions of n into 2 distinct coprime squares.
All such sublattices (including non-primitive ones) are counted in A025441.
The primitive sublattices that have the same symmetries (including the orientation of the mirrors) as the parent lattice are not counted here; they are counted in A019590, and all such sublattices (including non-primitive ones) are counted in A053866.
For n > 2, equals A193138. (End)

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. A193138, A145393 (all sublattices of the square lattice), A025441, A019590, A053866, A157226, A157230, A157231, A000089, A304182, A224450, A224770, A281877, A024362.

a(n) = (A000089(n) - A019590(n)) / 2. - Andrey Zabolotskiy, May 09 2018
a(n) = 1 if n>2 is in A224450, a(n) = 2 if n is in A224770, a(n) is a higher power of 2 if n is in A281877 (first time reaches 8 at n = 32045). - Andrey Zabolotskiy, Sep 30 2018
a(n) = b(n) for odd n, a(n) = b(n/2) for even n, where b(n) = A024362(n). - Andrey Zabolotskiy, Jan 23 2022

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

A193140 Number of isonemal satins of exact period n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 3, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 0, 3, 0, 1, 0, 3, 3, 0, 1, 3, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 0, 3, 3, 1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 0, 1, 7

Offset: 2

N. J. A. Sloane, Jul 16 2011

nonn

On page 153 of Grünbaum and Shephard (1980) is Table 3 which is a list of all the (n,s)-satins with n<=100. - Michael Somos, Dec 05 2014

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

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, page 152.

Cf. A193138, A193139, A086669, A000046, A005441, A262589.

#A193138
U:=proc(n) local j,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;
j:=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^(j-1)); fi;
end;
#A193139:
V:=proc(n) local j,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;
j:=t2-even;
if (al <= 1) then RETURN(2^(j-1)-1); fi;
if (al = 2) then RETURN(2^j-1); fi;
if (al >= 3) then RETURN(2^(j+1)-1); fi;
end;
#A193140:
[seq(U(n)+V(n), n=3..120)];

a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[
  f[[1, 1]] == 2 && f[[1, 2]] > 1,
  Boole[f[[1, 2]] == 2],
  Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]
]] - 1;
Table[a[n], {n, 2, 100}]
(* Andrey Zabolotskiy, Mar 21 2021 *)

a(n) = A086669(n) - 1. - Andrey Zabolotskiy, Dec 25 2018

a(2) = 0 prepended and name edited by Andrey Zabolotskiy, Mar 21 2021

A224770 Numbers that are the primitive sum of two squares in exactly two ways.

Original entry on oeis.org

65, 85, 130, 145, 170, 185, 205, 221, 265, 290, 305, 325, 365, 370, 377, 410, 425, 442, 445, 481, 485, 493, 505, 530, 533, 545, 565, 610, 629, 650, 685, 689, 697, 725, 730, 745, 754, 785, 793, 845, 850, 865, 890, 901, 905, 925, 949, 962, 965, 970

Offset: 1

Wolfdieter Lang, Apr 18 2013

nonn

These are the increasingly ordered numbers a(n) which satisfy A193138(a(n)) = 2.
Neither the order of the squares nor the signs of the numbers to be squared are taken into account. The two squares are necessarily distinct and each is nonzero.
This sequence is a proper subsequence of A000404.

			n=1,   65:  (1, 8),  (4, 7),
n=2,   85:  (2, 9),  (6, 7),
n=3,  130:  (3, 11), (7, 9),
n=4,  145:  (1, 12), (8, 9),
n=5,  170:  (1, 13), (7, 11),
n=6,  185:  (4, 13), (8, 11),
n=7,  205:  (3, 14), (6, 13),
n=8,  221:  (5, 14), (10, 11),
n=9,  265:  (3, 16), (11, 12),
n=10, 290:  (1, 17), (11, 13).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A224450 (one way), A193138 (multiplicities), A000404, A024509.

nn = 35; t = Sort[Select[Flatten[Table[If[GCD[a, b] == 1, a^2 + b^2, 0], {a, nn}, {b, a, nn}]], 0 < # <= nn^2 &]]; Transpose[Select[Tally[t], #[[2]] == 2 &]][[1]] (* T. D. Noe, Apr 20 2013 *)

a(n) = a^2 + b^2, a and integers, 0 < a < b and gcd(a,b) = 1 in exactly two ways. These representations of a(n) are denoted by two different pairs (a,b).

A224450 Numbers that are the primitive sum of two nonzero squares in exactly one way.

Original entry on oeis.org

2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 73, 74, 82, 89, 97, 101, 106, 109, 113, 122, 125, 137, 146, 149, 157, 169, 173, 178, 181, 193, 194, 197, 202, 218, 226, 229, 233, 241, 250, 257, 269, 274, 277, 281, 289, 293, 298, 313, 314, 317, 337

Offset: 1

Wolfdieter Lang, Apr 17 2013

nonn

If one includes 1 as the first entry then this sequence gives the numbers that are the primitive sum of two squares (square 0 allowed) in exactly one way, if neither the order of the squares nor the signs of the numbers to be squared matters.
Compare this sequence with A025284.
If 2 is omitted from this sequence then all members are primitively represented by two distinct nonzero squares in exactly one way.
The sequence A193138(n), n >= 3, gives the multiplicities of the primitive sums of two squares (automatically distinct and nonzero for n >= 3 if such a sum exists at all).
Numbers such that there is exactly one pair (m,k) where m + k = a(n), and m*k == 1 (mod a(n)), m > 0 and m <= k. - Torlach Rush, Oct 19 2020
A pair (s,t) such that s+t = a(n) and s*t == +1 (mod a(n)) as above is obtained from a square root of -1 (mod a(n)) for s and t = a(n)-s. - Joerg Arndt, Oct 24 2020

			a(1) = 2 because m = 2 is the first number with a unique doublet (a,b) in question, namely (1,1) (gcd(1,1) = 1).
This is the only case with equal entries a and b (the non-distinct case).
8 is not a member of this sequence (but of A025284) because the only representation is 2^2 +2^2 and (2,2) is not primitive. Similarly for 18, 20, ...
a(2) = 5 because 5 is the second smallest number satisfying the given requirements. 3 and 4 have no representation as sum of two nonzero squares, and the unique doublet for 5 is (1,2) (with distinct a and b).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A025284, A008784 (primitive sums of two squares with square 0 included), A224770 (exactly 2 ways), A193138 (multiplicities).

nn = 20; t = Sort[Select[Flatten[Table[If[GCD[a, b] == 1, a^2 + b^2, 0], {a, nn}, {b, a, nn}]], 0 < # <= nn^2 &]]; t2 = Transpose[Select[Tally[t], #[[2]] == 1 &]][[1]] (* T. D. Noe, Apr 20 2013 *)

This sequence gives the increasingly ordered numbers m which satisfy m = a^2 + b^2, with a and b integers, 0 < a <= b, gcd(a,b) = 1, and there is only one such representation, denoted by one doublet (a,b).

A106594 a(n) = number of primitive solutions to 4n+1 = x^2 + y^2.

Original entry on oeis.org

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

Offset: 1

Colin Mallows, May 10 2005

"Primitive" means that x and y are positive and mutually prime.

			E.g. a(16)=2 because 65 = 8^2+1^2 = 7^2+4^2. a(11)=0 because although 45=6^2+3^2, 6 and 3 are not mutually prime. a(2)=0 because although 9=3^2+0^2, 0 is not positive.

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A106602, A193138.

A106594 := proc(n)
      local a,x,y,fourn;
    fourn := 4*n+1 ;
    a := 0 ;
    for x from 1 do
        if x^2 >= fourn then
            return a;
        else
            y := fourn-x^2 ;
            if issqr(y) then
                y := sqrt(y) ;
                if y <= x and igcd(x,y) = 1 then
                    a := a+1 ;
                end if;
            end if;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 21 2013

Table[Length[If[CoprimeQ[#[[1]],#[[2]]],#,Nothing]&/@Union[Sort/@ ({#[[1,2]],#[[2,2]]}&/@FindInstance[{4 n+1==x^2+y^2,x>0,y>0},{x,y}, Integers,10])]],{n,100}] (* Harvey P. Dale, Jun 29 2021 *)

A193139 Number of symmetric satins of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 3, 0, 0, 1, 1, 1, 1, 0, 3, 0, 0, 0, 3, 1, 0, 1, 3, 0, 1, 1, 1, 1, 0, 1, 3, 0, 0, 1, 1, 0, 1, 0, 3, 3, 0, 0, 1, 0, 1, 1, 3, 0, 1, 1, 1, 1, 0, 1, 7

Offset: 3

N. J. A. Sloane, Jul 16 2011

nonn

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.

Cf. A193138, A193140, A157230.

V:=proc(n) local j,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;
j:=t2-even;
if (al <= 1) then RETURN(2^(j-1)-1); fi;
if (al = 2) then RETURN(2^j-1); fi;
if (al >= 3) then RETURN(2^(j+1)-1); fi;
end;
[seq(V(n),n=3..120)];

a(n) = A157230(n) - 1. - Andrey Zabolotskiy, Dec 25 2018

cp's OEIS Frontend

A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

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A193140 Number of isonemal satins of exact period n.

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A224770 Numbers that are the primitive sum of two squares in exactly two ways.

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A224450 Numbers that are the primitive sum of two nonzero squares in exactly one way.

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A106594 a(n) = number of primitive solutions to 4n+1 = x^2 + y^2.

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Maple

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A193139 Number of symmetric satins of order n.

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