A002747 Logarithmic numbers.

1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Comments

abs(a(n)) is also the number of distinct routes starting from a point A and ending at a point B, without traversing any edge more than once, when there are n bi-directional edges connecting A and B. E.g., if there are 3 edges p, q and r from A to B, then the 9 routes starting from A and ending at B are p, q, r, pqr, prq, rpq, rqp, qpr and qrp. - Nikita Kiran, Sep 02 2022

Reducing the sequence modulo the odd integer 2*k + 1 results in a purely periodic sequence with period dividing 4*k + 2, For example, reduced modulo 5 the sequence becomes the purely periodic sequence [1, 3, 4, 2, 0, 4, 2, 1, 3, 0, 1, 3, 4, 2, 0, 4, 2, 1, 3, 0, ...] with period 10. - Peter Bala, Sep 12 2022

References

• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
• Simon Plouffe, Simple inverter lookup on 1.8134302039235
• Simon Plouffe, Smart inverter lookup on 1.8134302039235
• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A000166, A000522, A087208.

Programs

Maple

a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2013

Mathematica

egf = x/Exp[x]/(1-x^2); a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)
a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 1, 22}] (* Peter Luschny, Dec 18 2017 *)

PARI

a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022

Formula

E.g.f.: x/exp(x)/(1-x^2). - Vladeta Jovovic, Feb 09 2003

a(n) = n*((n-1)*a(n-2)-(-1)^n). - Matthew Vandermast, Jun 30 2003

From Gerald McGarvey, Jun 06 2004: (Start)

For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.

For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.

For n odd, lim_{n->infinity} a(n)/n! = cosh(1).

For n even, lim_{n->infinity} a(n)/n! = sinh(1).

For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).

For signed values, Sum_{n>=1} a(n)/n!^2 = 0.

For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)

a(n) = (-1)^(n-1)*Sum_{k=0..n} C(n, k)*k!*(1-(-1)^k)/2. - Paul Barry, Sep 14 2004

a(n) = (-1)^(n+1)*n*A087208(n-1). - R. J. Mathar, Jul 24 2015

a(n) = (exp(-1)*Gamma(1+n,-1) - (-1)^n*exp(1)*Gamma(1+n,1))/2 = (A000166(n) - (-1)^n*A000522(n))/2. - Peter Luschny, Dec 18 2017

Extensions

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003