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a(n) = 10 + floor( (1 + Sum_{j=1..n-1} a(j) )/3 ).
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%I #6 Jul 07 2023 05:45:54

%S 10,13,18,24,32,42,56,75,100,133,178,237,316,421,562,749,999,1332,

%T 1776,2368,3157,4209,5612,7483,9977,13303,17737,23650,31533,42044,

%U 56059,74745,99660,132880,177173,236231,314975,419966,559955,746607

%N a(n) = 10 + floor( (1 + Sum_{j=1..n-1} a(j) )/3 ).

%H G. C. Greubel, <a href="/A120155/b120155.txt">Table of n, a(n) for n = 1..1000</a>

%t A120155[n_]:= A120155[n]= 10 +Quotient[1 +Sum[A120155[k], {k,n-1}], 3];

%t Table[A120155[n], {n,60}] (* _G. C. Greubel_, Jun 20 2023 *)

%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)/3);

%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 A120155:= func< n | g(n,10,1) >;

%o [A120155(n): n in [1..60]]; // _G. C. Greubel_, Jun 20 2023

%o (SageMath)

%o @CachedFunction

%o def A120155(n): return 10 +(1+sum(A120155(k) for k in range(1,n)))//3

%o [A120155(n) for n in range(1,61)] # _G. C. Greubel_, Jun 20 2023

%Y Cf. A072493, A073941, A112088.

%K nonn

%O 1,1

%A _Graeme McRae_, Jun 10 2006