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A309104
a(n) = Sum_{k >= 0} floor(n^(2*k+1) / (2*k+1)!).
2
0, 1, 3, 9, 25, 72, 199, 545, 1487, 4048, 11007, 29930, 81371, 221199, 601295, 1634499, 4443044, 12077466, 32829974, 89241138, 242582585, 659407853, 1792456409, 4872401708, 13244561050, 36002449653, 97864804699, 266024120286, 723128532126, 1965667148555
OFFSET
0,3
COMMENTS
This sequence is inspired by the Maclaurin series for the hyperbolic sine function.
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
See A309087 for similar sequences.
Cf. A000471.
Sequence in context: A291683 A079857 A265940 * A211282 A211298 A138574
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jul 12 2019
EXTENSIONS
Definition corrected by Robert Israel, Mar 18 2020
STATUS
approved