A174796 - OEIS

%I #25 Oct 11 2015 04:44:55

%S 1,2,7,30,143,728,3148,15986,86009,478907,2731365,13131703,72135374,

%T 412835191,2416852480,14369476066,72067537808,409636973141,

%U 2412844770335,14479410843183,87964452906330,451313038006432

%N Number of admissible sequences of order j; related to 7x+1 problem

%F A sequence s(k), where k=1, 2, ..., n, is *admissible* if it satisfies s(k)=7/2 exactly j times, s(k)=1/2 exactly n-j times, s(1)*s(2)*...*s(n) < 1 but s(1)*s(2)*...*s(m) > 1 for all 1 < m < n.

%F a(1)=1 and a(k+1)=Sum_{m=1..k} (-1)^(m-1)*binomial( floor( (k-m+1)*(log(7)/log(2)))+m-1,m)*a(k-m+1) ) for k >= 1. - _Vladimir M. Zarubin_, Sep 25 2015

%e The unique admissible sequence of order 1 is 7/2, 1/2, 1/2.

%e The two admissible sequences of order 2 are 7/2, 7/2, 1/2, 1/2, 1/2, 1/2 and 7/2, 1/2, 7/2, 1/2, 1/2, 1/2.

%t h[n_] := Module[{L = {{1}}}, For[i = 1, i <= n, i++, K = {}; S = 0; j = 1; While[7^i >= 2^(i + j - 1), If[7^(i - 1) >= 2^(i + j - 2), S = S + L[[i, j]]]; AppendTo[K, S]; j = j + 1]; AppendTo[L, K]; ]; Return[Map[Last, Drop[L, 1]]]]

%o (PARI)

%o n=20; a=vector(n); log72=log(7)/log(2);

%o {a[1]=1; for ( k=1, n-1, a[k+1]=sum( m=1,k,(-1)^(m-1)*binomial( floor( (k-m+1)*log72)+m-1,m)*a[k-m+1] ); print1(a[k], ", ") );} \\ _Vladimir M. Zarubin_, Sep 25 2015

%Y Similar to A100982 and A174795, but for the 7x+1 problem.

%K nonn

%O 1,2

%A T.M.M. Laarhoven (t.laarhoven(AT)gmail.com), Mar 29 2010