A309105 a(n) = Sum_{k >= 0} floor(n^(2*k) / (2*k)!).

1, 1, 3, 9, 25, 71, 198, 543, 1486, 4045, 11007, 29931, 81371, 221197, 601294, 1634497, 4443046, 12077467, 32829975, 89241140, 242582583, 659407855, 1792456409, 4872401706, 13244561047, 36002449653, 97864804698, 266024120284, 723128532126, 1965667148553

Offset: 0

Author

Rémy Sigrist, Jul 12 2019

Keywords

• nonn

Comments

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

Examples

			For n = 5:
- we have:
  k   5^(2*k)/(2*k)!
  --  --------------
   0               1
   1              12
   2              26
   3              21
   4               9
   5               2
   6               0
- hence a(5) = 1 + 12 + 26 + 21 + 9 + 2 = 71.
		

Links

• Wikipedia, Taylor series: Hyperbolic functions

Crossrefs

See A309087 for similar sequences.

Cf. A000501.

Programs

PARI

a(n) = { my (v=0, d=1); forstep (k=1, oo, 2, if (d<1, return (v), v += floor(d); d *= n^2/(k*(k+1)))) }

Formula

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

a(n) <= A000501(n).

Extensions

Definition corrected by Rémy Sigrist, Aug 06 2020