A065095 a(1) = 1, a(n+1) is the sum of a(n) and ceiling( arithmetic mean of a(1) ... a(n) ).

1, 2, 4, 7, 11, 16, 23, 33, 46, 62, 83, 110, 144, 186, 238, 303, 383, 481, 600, 744, 918, 1128, 1380, 1681, 2039, 2464, 2968, 3563, 4264, 5088, 6054, 7184, 8503, 10040, 11827, 13901, 16304, 19082, 22289, 25986, 30240, 35128, 40736, 47161, 54512

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 = 0.78... 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 = 0.2214496835182522607818590241239262909281832289078... It follows that the constant above is equal to C = 0.78501868866746800511978860290796656518270697588... - Vaclav Kotesovec, Oct 12 2024

Examples

			a(5) = a(4) + ceiling((a(1)+a(2)+a(3)+a(4))/4) = 7 + ceiling((1+2+4+7)/4) = 7 + floor(14/4) = 7 + 4 = 11.
		

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. A065094, A376995.

Programs

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] + ceil(summe/j): flip := 1-flip: od:

Mathematica

a[1] = 1; a[n_] := a[n] = a[n - 1] + Ceiling[ Sum[ a[i], {i, 1, n - 1} ]/(n - 1) ]; Table[ a[ n], {n, 1, 45} ]
Nest[Append[#,Last[#]+Ceiling[Mean[#]]]&,{1},44] (* James C. McMahon, Oct 10 2024 *)

PARI

{ for (n=1, 1000, if (n==1, s=0; a=1, s+=a; a+=ceil(s/(n - 1))); write("b065095.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

Formula

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