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For definition see comments lines.
0

%I #10 Dec 10 2016 03:12:58

%S 1,2,1,5,2,2,1,20,2,5,2,5,2,2,1,95,2,5,2,20,2,5,2,20,2,5,2,5,2,2,1,

%T 470,2,5,2,20,2,5,2,95,2,5,2,20,2,5,2,95,2,5,2,20,2,5,2,20,2,5,2,5,2,

%U 2,1,2345,2,5,2,20,2,5,2,95,2,5,2,20,2,5,2,470,2,5,2,20,2,5,2,95,2,5,2,20,2

%N For definition see comments lines.

%C It is easier to explain the rule of recurrence when the numbers are written as follows:

%C 1,

%C 2, 1,

%C 5, 2, 2, 1,

%C 20, 2, 5, 2, 5, 2, 2, 1,

%C 95, 2, 5, 2, 20, 2, 5, 2, 20, 2, 5, 2, 5, 2, 2, 1,

%C 470, 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 20, 2, 5, 2, 5, 2, 2, 1,

%C 2345, 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 470, 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 470,

%C 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 20, 2, 5, 2, 5, 2, 2, 1.

%C At first a(2^(n+1)-1) = (3*5^n+5)/4 (n>=0). Let A be the sequence defined as follows:

%C A(0)=2; W(A(0))=5; A(1)=A(0),W(A(0))=2, 5; W(A(1))=2, 20.

%C More generally with A(n)=B(n), {3*5^n+5)/4; we define W(A(n))=B(n), (3*5^(n+1)+5)/4 and A(n+1)=A(n), W(A(n)).

%C Here we obtain A(1)=2, 5; W(A(1))=2, 20; A(2)=2, 5, 2, 20; W(A(2))=2, 5, 2, 95; A(3)=2, 5, 2, 20, 2, 5, 2, 95;

%C W(A(3))=2, 5, 2, 20, 2, 5, 2, 470; A(4)=2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, 470, etc.

%C In fact: B(1)=2; B(2)=2, 5, 2; B(3)=2, 5, 2, 20, 2, 5, 2; B(4)=2, 5, 2, 20, 2, 5, 2, 95, 2, 5, 2, 20, 2, 5, 2, etc.

%C If we denote by <<A|UA|>> the subsequence of a between a(2^(n+1)-1) and a(2^(n+2)-1), the subsequence of a between a(2^(n+2)-1) and a(2^(n+3)-1) is given by <<A|A(n+1), A(n+1), UA|>>.

%C It seems that this sequence gives the numbers of 1 in the successive sets of 1 in the sequence A174835.

%e a(1)=a(2^1-1)=(3*5^0+5)/4=2. a(3)=a(2^2-1)=(3*5+5)/4=5.

%e a(7)=a(2^3-1)=(75+5)/4=20. a(15)=a(2^4-1)=(3*125+5)/4=380/4=95.

%e Between 20 and 95 the subsequence of a is: 2, 5, 2, 5, 2, 2, 1.

%e Then with the definition, the subsequence of a, between 95 and 470 is:

%e A(2), A(2), 2, 5, 2, 5, 2, 2, 1, i.e., 2, 5, 2, 20, 2, 5, 2, 20, 2, 5, 2, 5, 2, 2, 1.

%Y Cf. A174835, A174837.

%K easy,nonn,uned

%O 0,2

%A _Richard Choulet_, Apr 03 2010