login
a(1) = 1, a(n+1) is the sum of a(n) and floor( arithmetic mean of a(1) ... a(n) ).
11

%I #19 Oct 12 2024 10:49:08

%S 1,2,3,5,7,10,14,20,27,36,48,63,82,106,136,173,218,273,341,423,522,

%T 641,784,955,1158,1399,1685,2023,2421,2889,3437,4079,4828,5701,6716,

%U 7893,9257,10834,12655,14754,17169,19944,23128,26775,30948,35716,41157

%N a(1) = 1, a(n+1) is the sum of a(n) and floor( arithmetic mean of a(1) ... a(n) ).

%C It seems that a(n) is asymptotic to C*BesselI(0,2*sqrt(n)) where C is a constant C = 0.44... and BesselI(b,x) is the modified Bessel function of the first kind. Can someone prove this?

%C Numerically, a(n) ~ c * exp(2*sqrt(n)) / n^(1/4), where c = 0.12571987512700920098166979884420897638511306007242... It follows that the constant above is equal to C = 0.445665353608456118285630970456186510059368576678... - _Vaclav Kotesovec_, Oct 12 2024

%C A241772(n) = a(n+1) - a(n) = (Sum_{1..n} a(k)) / n. - _Reinhard Zumkeller_, Apr 28 2014

%H Harry J. Smith, <a href="/A065094/b065094.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F a(1) = 1, a(n+1) = a(n) + floor((a(1) + a(2) + ... + a(n))/n).

%e a(5) = a(4) + floor((a(1)+a(2)+a(3)+a(4))/4) = 5 + floor((1+2+3+5)/4) = 5 + floor(11/4) = 5 + 2 = 7.

%p a[1] := 1: summe := 0: flip := 1: for j from 1 to 100 do: print (j, a[flip]); summe := summe + a[flip]: a[1-flip] := a[flip] + floor(summe/j): flip := 1-flip: od:

%t a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[ Sum[ a[i], {i, 1, n - 1} ]/(n - 1) ]; Table[ a[n], {n, 1, 47} ]

%t Nest[Append[#, Last[#]+Floor[Mean[#]]]&, {1}, 46] (* _James C. McMahon_, Oct 11 2024 *)

%o (PARI) { for (n=1, 1000, if (n==1, s=0; a=1, s+=a; a+=s\(n - 1)); write("b065094.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 06 2009

%o (Haskell)

%o a065094 n = a065094_list !! (n-1)

%o a065094_list = 1 : f 1 1 1 where

%o f k s x = y : f (k + 1) (s + y) y where y = x + div s k

%o -- _Reinhard Zumkeller_, Apr 28 2014

%Y Cf. A065095, A376995.

%K nonn,easy

%O 1,2

%A Ulrich Schimke (ulrschimke(AT)aol.com)