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

%I A065095 #18 Oct 12 2024 11:47:12
%S A065095 1,2,4,7,11,16,23,33,46,62,83,110,144,186,238,303,383,481,600,744,918,
%T A065095 1128,1380,1681,2039,2464,2968,3563,4264,5088,6054,7184,8503,10040,
%U A065095 11827,13901,16304,19082,22289,25986,30240,35128,40736,47161,54512
%N A065095 a(1) = 1, a(n+1) is the sum of a(n) and ceiling( arithmetic mean of a(1) ... a(n) ).
%C A065095 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?
%C A065095 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
%H A065095 Harry J. Smith, <a href="/A065095/b065095.txt">Table of n, a(n) for n=1,...,1000</a>
%H A065095 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A065095 a(1) = 1, a(n+1) = a(n) + ceiling((a(1) + a(2) + ... + a(n))/n).
%e A065095 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.
%p A065095 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:
%t A065095 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} ]
%t A065095 Nest[Append[#,Last[#]+Ceiling[Mean[#]]]&,{1},44] (* _James C. McMahon_, Oct 10 2024 *)
%o A065095 (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
%Y A065095 Cf. A065094, A376995.
%K A065095 nonn,easy
%O A065095 1,2
%A A065095 Ulrich Schimke (ulrschimke(AT)aol.com)