A358987 Omit the trailing 5 from double factorial of odd numbers (A001147(n)).

1, 1, 3, 1, 10, 94, 1039, 13513, 202702, 3445942, 65472907, 1374931057, 31623414322, 790585358062, 21345804667687, 619028335362937, 19189878396251062, 633265987076285062, 22164309547669977187, 820079453263789155937, 31983098677287777081562, 1311307045768798860344062

Offset

0,3

Comments

A001147(n) has only a trailing five for n > 2.

Proof: being trivial to prove that A001147(n) ends with at least a digit 5 for n > 2, it remains to prove that the tenth digit of A001147(n) is not equal to 5. Considering the product A001147(n) = A001147(n-1)*(2*n - 1) for n > 2, it is easy to verify that the tenth digit of A001147(n) is congruent to 2 modulo 5 if the tenth digit of A001147(n-1) is congruent to 2 modulo 5. Since for n <= 8 the tenth digit of A001147(n) is not equal to 5, and it is equal to 2 for n = 8, it follows that the tenth digit of A001147(n) for n > 8 is congruent to 2 modulo 5, and therefore, not equal to 5. QED.

Links

Stefano Spezia, Table of n, a(n) for n = 0..400

Formula

a(n) = (A001147(n) - 5)/10 for n > 2.

E.g.f.: (14 + 2*x*(7 + 8*x) - 5*exp(x) + 1/sqrt(1 - 2*x))/10.

D-finite with recurrence a(n) + (-2*n+1)*a(n-1) + (-n+1) = 0 for n > 3. - R. J. Mathar, Mar 25 2024

Mathematica

Join[{1, 1, 3}, Table[((2n-1)!!-5)/10, {n, 3, 21}]] (* or *)
CoefficientList[Series[(14-5Exp[x]+1/Sqrt[1-2x]+2x(7+8x))/10, {x, 0, 21}], x]Table[n!, {n, 0, 21}]

Crossrefs

Cf. A001147.

Sequence in context: A225753 A210725 A048953 * A200652 A276391 A119632

Adjacent sequences: A358984 A358985 A358986 * A358988 A358989 A358990

Keyword

nonn,base,easy

Author

Stefano Spezia, Dec 10 2022

Status

approved