

A193138


Number of square satins of order n.


6



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
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OFFSET

3,63


COMMENTS

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 NivenZuckermanMontgomery 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 GrünbaumShephard 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 allblack 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


REFERENCES

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.


LINKS

Table of n, a(n) for n=3..120.
B. Grünbaum and G. C. Shephard, Satins and twills: an introduction to the geometry of fabrics, Math. Mag., 53 (1980), 139161. See Theorem 5, p. 152.


FORMULA

Take the prime number factorization (symbolically) as n = 2^a*product(p^b)*product(p^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^(t1) with t the number of distinct primes congruent 1(mod 4). Otherwise a(n) = 0. (See the NivenZuckermanMontgomery reference, Theorem 3.22, p. 165, and the GrünbaumShephard Theorem 5 formula for u(n)).  Wolfdieter Lang, Apr 18 2013


EXAMPLE

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


MAPLE

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:=t2even;
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^(nop1)); fi;
end;
[seq(U(n), n=3..120)];


MATHEMATICA

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


CROSSREFS

Cf. A193139, A193140, A157228.
Sequence in context: A322583 A102354 A157228 * A255320 A256574 A304819
Adjacent sequences: A193135 A193136 A193137 * A193139 A193140 A193141


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jul 16 2011


STATUS

approved



