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A174796 Number of admissible sequences of order j; related to 7x+1 problem.

%I A174796 #31 Jul 22 2025 07:54:56
%S A174796 1,2,7,30,143,728,3148,15986,86009,478907,2731365,13131703,72135374,
%T A174796 412835191,2416852480,14369476066,72067537808,409636973141,
%U A174796 2412844770335,14479410843183,87964452906330,451313038006432
%N A174796 Number of admissible sequences of order j; related to 7x+1 problem.
%F A174796 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 A174796 a(1) = 1 and a(k) = Sum_{m=1..k-1} (-1)^(m-1)*binomial(floor((k-m)*(log(7)/log(2)))+m-1, m)*a(k-m) for k >= 2. - _Vladimir M. Zarubin_, Sep 25 2015
%e A174796 The unique admissible sequence of order 1 is 7/2, 1/2, 1/2.
%e A174796 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 A174796 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 A174796 (PARI)
%o A174796 n=20; a=vector(n); log72=log(7)/log(2);
%o A174796 {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 A174796 Similar to A100982 and A174795, but for the 7x+1 problem.
%K A174796 nonn
%O A174796 1,2
%A A174796 T.M.M. Laarhoven (t.laarhoven(AT)gmail.com), Mar 29 2010