A234717 a(n) = floor(n/(exp(1/(2*n))-1)).

1, 7, 16, 30, 47, 69, 94, 124, 157, 195, 236, 282, 331, 385, 442, 504, 569, 639, 712, 790, 871, 957, 1046, 1140, 1237, 1339, 1444, 1554, 1667, 1785, 1906, 2032, 2161, 2295, 2432, 2574, 2719, 2869, 3022, 3180, 3341, 3507

Offset: 1

Author

Richard R. Forberg, Dec 29 2013

Keywords

• nonn
• easy

Comments

Equations of this general form: (n/(exp(1/(r*n))-1)) have a fractional portion that converges to one or more rational fractions if r is rational. They have second differences that are nearly constant before the floor function, and repeat in patterns when calculated after the floor function.

The fractional portion of this equation (before the floor function) oscillates between two fractions that converge towards 1/24 and 13/24.

Second differences of a(n) = repeat{3,5}.

First differences of a(n) = A075123(n+3).

Partial sums of a(n) = A033951(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A033951, A075123.

Programs

Mathematica

Table[Floor[n/(Exp[1/(2 n)] - 1)], {n, 100}] (* Wesley Ivan Hurt, Apr 01 2022 *)

Formula

From Ralf Stephan, Mar 28 2014: (Start)

a(n) = (1/4)*(8n^2 - 2n - 1 + (-1)^n).

G.f.: x*(2*x^2 + 5*x + 1)/((1-x^2)*(1-x)^2). (End)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Wesley Ivan Hurt, Apr 01 2022

E.g.f.: (x*(4*x + 3)*cosh(x) + (4*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Nov 23 2023