A065094 a(1) = 1, a(n+1) is the sum of a(n) and floor( arithmetic mean of a(1) ... a(n) ).

1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 106, 136, 173, 218, 273, 341, 423, 522, 641, 784, 955, 1158, 1399, 1685, 2023, 2421, 2889, 3437, 4079, 4828, 5701, 6716, 7893, 9257, 10834, 12655, 14754, 17169, 19944, 23128, 26775, 30948, 35716, 41157

Offset: 1

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• easy

Comments

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?

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

A241772(n) = a(n+1) - a(n) = (Sum_{1..n} a(k)) / n. - Reinhard Zumkeller, Apr 28 2014

Examples

			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.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A065095, A376995.

Programs

Haskell

a065094 n = a065094_list !! (n-1)
a065094_list = 1 : f 1 1 1 where
   f k s x = y : f (k + 1) (s + y) y where y = x + div s k
-- Reinhard Zumkeller, Apr 28 2014

Maple

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:

Mathematica

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} ]
Nest[Append[#, Last[#]+Floor[Mean[#]]]&, {1}, 46] (* James C. McMahon, Oct 11 2024 *)

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

Formula

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