A078141 a(n) = floor(exp(n - gamma)), where gamma is Euler's constant.

0, 1, 4, 11, 30, 83, 226, 615, 1673, 4549, 12366, 33616, 91380, 248397, 675213, 1835420, 4989191, 13562027, 36865412, 100210580, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245428

Offset: 0

Author

Michael Somos, Nov 20 2002

Keywords

• nonn
• easy

Comments

For any fixed real value x > 0 let u(1) = 1 and u(n) = u(n-1) - x*sign(u(n-1))/n. Then the series S(x) = Sum_{k>=1} u(k) converges and has value S(x) = x*floor(exp(1/x + gamma - 1)) - x - 1 where gamma = 0.5772... is the Euler gamma constant. Thus for n >= 0 a(n+1) = n*S(1/n) + n + 1.

From Gerhard Kirchner, May 02 2020: (Start)

It is interesting to compare T(x)=exp(x-gamma) and G(x)=[T(x)] with F(x)= index k of the harmonic sequence H(k) such that H(k) is closest to x. More exactly: For H(k-1)=x1, where x1=H(k)-1/(2k) is the point of dicontinuity. Harmonic sequence: H(0)=0 and H(k)=H(k-1)+1/k for k>0 with the asymptotic formula H(k)=log(k)+gamma+1/(2k)-1/(12k^2)+O(1/k^4).

G(x) has a point of dicontinuity x2=x1+1/(12k^2), which can be derived from T(x2)=k and the asymptotic formula. Within the "critical" section x1

What about n>200? Let x1

Regarding the difference as a random number between 0 and 1 and considering the exponential increase of T(n), it is unlikely that any n is located in a critical section.

(End)

Crossrefs

Cf. A002387, A004080.

Programs

PARI

a(n)=if(n<0,0,floor(exp(n-Euler)))

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.