A065094 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

Index entries for sequences related to Bessel functions or polynomials

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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

(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

CROSSREFS

Cf. A065095, A376995.