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A002747
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Logarithmic numbers.
(Formerly M1924 N0759)
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5
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1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
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MATHEMATICA
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a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
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PROG
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(PARI) a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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