OFFSET
0,3
COMMENTS
This sequence is inspired by the Maclaurin series for the hyperbolic sine function.
LINKS
Robert Israel, Table of n, a(n) for n = 0..2300
Wikipedia, Taylor series: Hyperbolic functions
FORMULA
a(n) ~ sinh(n) as n tends to infinity.
a(n) <= A000471(n).
EXAMPLE
For n = 5:
- we have:
k 5^(2*k+1)/(2*k+1)!
- ------------------
0 5
1 20
2 26
3 15
4 5
5 1
>=6 0
- hence a(5) = 5 + 20 + 26 + 15 + 5 + 1 = 72.
MAPLE
f:= proc(n) local t, k, v;
v:= n; t:= n;
for k from 1 do
v:= v*n^2/(2*k*(2*k+1));
if v < 1 then return t fi;
t:= t + floor(v);
od
end proc:
map(f, [$0..30]); # Robert Israel, Mar 18 2020
PROG
(PARI) a(n) = { my (v=0, d=n); forstep (k=2, oo, 2, if (d<1, return (v), v += floor(d); d *= n^2/(k*(k+1)))) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jul 12 2019
EXTENSIONS
Definition corrected by Robert Israel, Mar 18 2020
STATUS
approved