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a(n) = a(n-2) + a(n-3) if n == 0 (mod 3), a(n-1) + a(n-4) if n == 0 (mod 4), otherwise a(n-2) with a(0) = 0 and a(1) = a(2) = a(3) = 1.
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%I #12 Mar 30 2021 05:21:00

%S 0,1,1,1,1,1,2,2,3,4,4,4,8,8,8,16,24,24,40,40,64,80,80,80,160,160,160,

%T 320,480,480,800,800,1280,1600,1600,1600,3200,3200,3200,6400,9600,

%U 9600,16000,16000,25600,32000,32000,32000,64000,64000,64000,128000,192000,192000,320000,320000,512000,640000,640000,640000,1280000

%N a(n) = a(n-2) + a(n-3) if n == 0 (mod 3), a(n-1) + a(n-4) if n == 0 (mod 4), otherwise a(n-2) with a(0) = 0 and a(1) = a(2) = a(3) = 1.

%C Lim_{n -> infinity} <a(n+1)/a(n)> = 1.324717957244746, where <> is the expectation value.

%D .

%H G. C. Greubel, <a href="/A141525/b141525.txt">Table of n, a(n) for n = 0..500</a>

%t a[n_]:= a[n]= If[n==0, 0, If[n<4, 1, If[Mod[n, 3]==0, a[n-2] + a[n-3], If[Mod[n, 4] ==0, a[n-1] + a[n-4], a[n-1] ]]]];

%t Table[a[n], {n, 0, 65}] (* modified by _G. C. Greubel_, Mar 29 2021 *)

%o (Magma)

%o function a(n)

%o if n eq 0 then return 0;

%o elif n lt 4 then return 1;

%o elif (n mod 3) eq 0 then return a(n-2) + a(n-3);

%o elif (n mod 4) eq 0 then return a(n-1) + a(n-4);

%o else return a(n-1);

%o end if; return a;

%o end function;

%o [a(n): n in [0..65]]; // _G. C. Greubel_, Mar 29 2021

%o (Sage)

%o @CachedFunction

%o def a(n):

%o if (n==0): return 0

%o elif (n<4): return 1

%o elif (n%3==0): return a(n-2) + a(n-3)

%o elif (n%4==0): return a(n-1) + a(n-4)

%o else: return a(n-1)

%o [a(n) for n in (0..65)] # _G. C. Greubel_, Mar 29 2021

%K nonn

%O 0,7

%A _Roger L. Bagula_, Aug 11 2008

%E Edited by _G. C. Greubel_, Mar 29 2021