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

1, 2, 7, 30, 143, 728, 3148, 15986, 86009, 478907, 2731365, 13131703, 72135374, 412835191, 2416852480, 14369476066, 72067537808, 409636973141, 2412844770335, 14479410843183, 87964452906330, 451313038006432

Offset

1,2

Links

Table of n, a(n) for n=1..22.

Formula

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.

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

Example

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

Mathematica

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]]]]

Prog

(PARI)
n=20; a=vector(n); log72=log(7)/log(2);
{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

Crossrefs

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

Sequence in context: A260771 A260773 A368937 * A046648 A006013 A358965

Adjacent sequences: A174793 A174794 A174795 * A174797 A174798 A174799

Keyword

nonn

Author

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

Status

approved