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

1,2

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.

Links

Table of n, a(n) for n=1..70.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A006882.

Sequence in context: A249167 A205101 A069752 * A238778 A004731 A135354

Adjacent sequences: A265691 A265692 A265693 * A265695 A265696 A265697

Keyword

nonn,easy

Author

Altug Alkan, Dec 13 2015

Status

approved