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a(n) = 3 + floor((2 + Sum_{j=1..n-1} a(j))/5).
1

%I #11 Dec 26 2023 11:23:24

%S 3,4,4,5,6,7,9,11,13,15,18,22,26,32,38,46,55,66,79,95,114,137,164,197,

%T 236,283,340,408,490,588,705,846,1015,1218,1462,1754,2105,2526,3031,

%U 3638,4365,5238,6286,7543,9052,10862,13034,15641,18769,22523,27028,32433

%N a(n) = 3 + floor((2 + Sum_{j=1..n-1} a(j))/5).

%H G. C. Greubel, <a href="/A120172/b120172.txt">Table of n, a(n) for n = 1..10000</a>

%t nxt[{a_,ls_}]:=Module[{x=Floor[(17+ls)/5]},{x,ls+x}]; Transpose[ NestList[ nxt,{3,3},60]][[1]] (* _Harvey P. Dale_, Jun 11 2014 *)

%o (Magma)

%o function f(n, a, b)

%o t:=0;

%o for k in [1..n-1] do

%o t+:= a+Floor((b+t)/5);

%o end for;

%o return t;

%o end function;

%o g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;

%o A120172:= func< n | g(n, 3, 2) >;

%o [A120172(n): n in [1..60]]; // _G. C. Greubel_, Dec 25 2023

%o (SageMath)

%o @CachedFunction

%o def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//5

%o def A120172(n): return f(n, 3, 2)

%o [A120172(n) for n in range(1, 61)] # _G. C. Greubel_, Dec 25 2023

%Y Cf. A073941, A072493, A112088.

%K nonn

%O 1,1

%A _Graeme McRae_, Jun 10 2006

%E More terms from _Harvey P. Dale_, Jun 11 2014