A309104 a(n) = Sum_{k >= 0} floor(n^(2*k+1) / (2*k+1)!).

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

Author

Rémy Sigrist, Jul 12 2019

Keywords

• nonn

Comments

This sequence is inspired by the Maclaurin series for the hyperbolic sine function.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2300
• Wikipedia, Taylor series: Hyperbolic functions

Crossrefs

See A309087 for similar sequences.

Cf. A000471.

Programs

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

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)))) }

Formula

a(n) ~ sinh(n) as n tends to infinity.

a(n) <= A000471(n).

Extensions

Definition corrected by Robert Israel, Mar 18 2020