1,3

Eric Roosendaal counted all admissible sequences up to order j=1000 (2005). Note: there is a typo in both Wagon and Chamberland in the definition of Wagon's constant 9.477955... The expression floor(1+2*i+i*log_2(3)) should be replaced by floor(1+i+i*log_2(3)).

The length of all admissible sequences of order j is A020914(j). - T. D. Noe (noe(AT)sspectra.com), Sep 11 2006

M. Chamberland, Una actualizacio del problema 3x+1, Butl. Soc. Catalana Mat. 18 (2003) 19-45.

S. Wagon, The Collatz problem, Math. Intelligencer 7 (1985) 72-76.

T. D. Noe, Table of n, a(n) for n=1..500

M. Chamberland, English translation.

A sequence s(k), where k=1, 2, ..., n, is *admissible* if it satisfies s(k)=3/2 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.

The unique admissible sequence of order 1 is 3/2, 1/2.

The unique admissible sequence of order 2 is 3/2, 3/2, 1/2, 1/2.

The two admissible sequences of order 3 are 3/2, 3/2, 3/2, 1/2, 1/2 and 3/2, 3/2, 1/2, 3/2, 1/2.

(* based on Eric Roosendaal's algorithm *) nn=100; Clear[x, y]; Do[x[i]=0, {i, 0, nn+1}]; x[1]=1; t=Table[Do[y[cnt]=x[cnt]+x[cnt-1], {cnt, p+1}]; Do[x[cnt]=y[cnt], {cnt, p+1}]; admis=0; Do[If[(p+1-cnt)*Log[3]<p*Log[2], admis=admis+x[cnt]; x[cnt]=0], {cnt, p+1}]; admis, {p, 2, nn}]; DeleteCases[t, 0] - T. D. Noe (noe(AT)sspectra.com), Sep 11 2006

Cf. A122790 (Wagon's constant).

Sequence in context: A111759 A047749 A134565 this_sequence A034786 A080107 A056156

Adjacent sequences: A100979 A100980 A100981 this_sequence A100983 A100984 A100985

nonn

S. R. Finch (Steven.Finch(AT)inria.fr), Jan 13 2005

Two more terms from Jules Renucci (jules.renucci(AT)wanadoo.fr), Nov 02 2005

More terms from T. D. Noe (noe(AT)sspectra.com), Sep 11 2006