A265694 a(n) = n!! mod n^2 where n!! is a double factorial number (A006882).

0, 2, 3, 8, 15, 12, 7, 0, 54, 40, 110, 0, 104, 84, 0, 0, 221, 0, 342, 0, 0, 220, 506, 0, 0, 312, 0, 0, 493, 0, 930, 0, 0, 544, 0, 0, 222, 684, 0, 0, 369, 0, 1806, 0, 0, 1012, 47, 0, 0, 0, 0, 0, 1590, 0, 0, 0, 0, 1624, 59, 0, 3050, 1860, 0, 0, 0, 0, 4422, 0, 0, 0

Offset: 1

Author

Altug Alkan, Dec 13 2015

Keywords

• nonn
• easy

Comments

Inspired by geometric meaning of distribution of 0's in this sequence.

Position of 0's in this sequence is directly related with sequence which gives the short leg of more than one Pythagorean triangle (A009188). See comment sections in A009188 and A264828 which are the related sequences for further information.

More precisely, a(A009188(n+1)) = 0 for n > 0.

Examples

			For n = 1, a(1) = 1!! mod 1^2 = 1 mod 1 = 0.
For n = 2, a(2) = 2!! mod 2^2 = 2 mod 4 = 2.
For n = 8, a(8) = 8!! mod 8^2 = 384 mod 64 = 0.
		

Crossrefs

Cf. A006882.

Programs

Magma

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n) mod n^2: n in [1..70] ]; // Vincenzo Librandi, Dec 14 2015

Mathematica

Table[Mod[n!!, n^2], {n, 70}] (* Michael De Vlieger, Dec 14 2015 *)

PARI

df(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n));
vector(70, n, df(n) % n^2)

Formula

a(A009188(n+1)) = 0 for n > 0.