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A120167 a(n) = 9 + floor((3 + Sum_{j=1..n-1} a(j))/4).

%I A120167 #10 Sep 10 2023 09:07:41
%S A120167 9,12,15,18,23,29,36,45,56,70,88,110,137,171,214,268,335,418,523,654,
%T A120167 817,1021,1277,1596,1995,2494,3117,3896,4870,6088,7610,9512,11890,
%U A120167 14863,18579,23223,29029,36286,45358,56697
%N A120167 a(n) = 9 + floor((3 + Sum_{j=1..n-1} a(j))/4).
%H A120167 Harvey P. Dale, <a href="/A120167/b120167.txt">Table of n, a(n) for n = 1..1000</a>
%t A120167 nxt[{t_,a_}]:=Module[{c=Floor[(39+t)/4]},{t+c,c}]; NestList[nxt,{9,9},40][[All,2]] (* _Harvey P. Dale_, Apr 24 2019 *)
%o A120167 (Magma)
%o A120167 function f(n, a, b)
%o A120167   t:=0;
%o A120167     for k in [1..n-1] do
%o A120167       t+:= a+Floor((b+t)/4);
%o A120167     end for;
%o A120167   return t;
%o A120167 end function;
%o A120167 g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
%o A120167 A120167:= func< n | g(n, 9, 3) >;
%o A120167 [A120167(n): n in [1..60]]; // _G. C. Greubel_, Sep 09 2023
%o A120167 (SageMath)
%o A120167 @CachedFunction
%o A120167 def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
%o A120167 def A120167(n): return f(n, 9, 3)
%o A120167 [A120167(n) for n in range(1, 61)] # _G. C. Greubel_, Sep 09 2023
%Y A120167 Cf. A072493, A073941, A112088.
%Y A120167 Cf. A120160 - A120166, A120168, A120169.
%K A120167 nonn,easy
%O A120167 1,1
%A A120167 _Graeme McRae_, Jun 10 2006