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

%I A309104 #10 Mar 18 2020 23:02:04
%S A309104 0,1,3,9,25,72,199,545,1487,4048,11007,29930,81371,221199,601295,
%T A309104 1634499,4443044,12077466,32829974,89241138,242582585,659407853,
%U A309104 1792456409,4872401708,13244561050,36002449653,97864804699,266024120286,723128532126,1965667148555
%N A309104 a(n) = Sum_{k >= 0} floor(n^(2*k+1) / (2*k+1)!).
%C A309104 This sequence is inspired by the Maclaurin series for the hyperbolic sine function.
%H A309104 Robert Israel, <a href="/A309104/b309104.txt">Table of n, a(n) for n = 0..2300</a>
%H A309104 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taylor_series#Hyperbolic_functions">Taylor series: Hyperbolic functions</a>
%F A309104 a(n) ~ sinh(n) as n tends to infinity.
%F A309104 a(n) <= A000471(n).
%e A309104 For n = 5:
%e A309104 - we have:
%e A309104   k  5^(2*k+1)/(2*k+1)!
%e A309104   -  ------------------
%e A309104   0                   5
%e A309104   1                  20
%e A309104   2                  26
%e A309104   3                  15
%e A309104   4                   5
%e A309104   5                   1
%e A309104   >=6                 0
%e A309104 - hence a(5) = 5 + 20 + 26 + 15 + 5 + 1 = 72.
%p A309104 f:= proc(n) local t,k,v;
%p A309104   v:= n; t:= n;
%p A309104   for k from 1 do
%p A309104     v:= v*n^2/(2*k*(2*k+1));
%p A309104     if v < 1 then return t fi;
%p A309104     t:= t + floor(v);
%p A309104   od
%p A309104 end proc:
%p A309104 map(f, [$0..30]); # _Robert Israel_, Mar 18 2020
%o A309104 (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)))) }
%Y A309104 See A309087 for similar sequences.
%Y A309104 Cf. A000471.
%K A309104 nonn
%O A309104 0,3
%A A309104 _Rémy Sigrist_, Jul 12 2019
%E A309104 Definition corrected by _Robert Israel_, Mar 18 2020