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A258875 a(1) = a(2) = a(3) = 1; for n > 3, a(n) = ceiling((a(n-1) + a(n-2) + a(n-3))/2).

%I A258875 #29 Sep 08 2022 08:46:12
%S A258875 1,1,1,2,2,3,4,5,6,8,10,12,15,19,23,29,36,44,55,68,84,104,128,158,195,
%T A258875 241,297,367,453,559,690,851,1050,1296,1599,1973,2434,3003,3705,4571,
%U A258875 5640,6958,8585,10592,13068,16123,19892,24542,30279
%N A258875 a(1) = a(2) = a(3) = 1; for n > 3, a(n) = ceiling((a(n-1) + a(n-2) + a(n-3))/2).
%C A258875 First 14 terms are the same as A179241.
%C A258875 Ratio of consecutive terms approaches the real root of x^3 - (x^2 + x + 1)/2 = 0, whose approximate value is 1.2337519285, and whose exact value is (1 + (64 - 3*sqrt(417))^(1/3) + (64 + 3*sqrt(417))^(1/3))/6.
%C A258875 Same as A180235 for n > 5. - _Georg Fischer_, Oct 09 2018
%H A258875 Vincenzo Librandi, <a href="/A258875/b258875.txt">Table of n, a(n) for n = 1..2000</a>
%p A258875 a(4) = ceiling((1+1+1)/2) = 2;
%p A258875 a(5) = ceiling((1+1+2)/2) = 2;
%p A258875 a(6) = ceiling((1+2+2)/2) = 3.
%t A258875 RecurrenceTable[{a[n] == Ceiling[(a[n - 1] + a[n - 2] + a[n - 3])/2], a[1] == a[2] == a[3] == 1}, a, {n, 1, 49}] (* _Michael De Vlieger_, Jun 20 2015 *)
%t A258875 nxt[{a_,b_,c_}]:={b,c,Ceiling[(a+b+c)/2]}; NestList[nxt,{1,1,1},50][[All,1]] (* _Harvey P. Dale_, Feb 03 2022 *)
%o A258875 (PARI) lista(nn) = {va = vector(nn, n, if (n<=3, 1)); for (n=4, nn, va[n] = ceil((va[n-1]+va[n-2]+va[n-3])/2);); va;} \\ _Michel Marcus_, Jun 17 2015
%o A258875 (Magma) [n le 3 select 1 else Ceiling((Self(n-1)+Self(n-2)+ Self(n-3))/2): n in [1..60]]; // _Vincenzo Librandi_, Oct 10 2018
%K A258875 nonn
%O A258875 1,4
%A A258875 _Morris Neene_, Jun 13 2015