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

1,2

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A065094, A376995.

Sequence in context: A000601 A062433 A317910 * A005253 A212364 A320591

Adjacent sequences: A065092 A065093 A065094 * A065096 A065097 A065098

Keyword

nonn,easy

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Status

approved