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%I #49 Jun 27 2020 05:10:52
%S 1,3,1,3,1,4,1,5,3,4,1,5,3,6,4,3,7,1,5,6,4,3,7,1,5,8,6,4,9,1,5,3,7,8,
%T 6,4,9,1,5,3,7,10,8,6,4,11,3,7,1,5,9,10,8,6,4,11,3,7,1,5,9,12,10,8,6,
%U 4,9,1,13,5,3,7,11,12,10,8,6,4,9,1,13,5,3,7,11,14,12,10,8,6,4,11,3,15
%N Triangle T(n,k) read by rows: in the Josephus problem with n initial numbers on a line: eliminate each second and reverse left-right-direction of elimination. T(n,k) is the (n-k+1)st element removed, 1<=k<=n.
%H Georg Fischer, <a href="/A335552/b335552.txt">Table of n, a(n) for n = 1..1000</a>
%H K. Matsumoto, T. Nakamigawa, M. Watanabe, <a href="http://hdl.handle.net/10131/5785">On the switchback vertion of Josephus Problem</a>, Yokohama Math. J. 53 (2007) 83, function f_k(n).
%H <a href="/index/J#nome">Index to sequences related to the Josephus problem</a>
%e The triangle starts
%e 1
%e 3 1
%e 3 1 4
%e 1 5 3 4
%e 1 5 3 6 4
%e 3 7 1 5 6 4
%e 3 7 1 5 8 6 4
%e 9 1 5 3 7 8 6 4
%e 9 1 5 3 7 10 8 6 4
%e 11 3 7 1 5 9 10 8 6 4
%e 11 3 7 1 5 9 12 10 8 6 4
%e 9 1 13 5 3 7 11 12 10 8 6 4
%e 9 1 13 5 3 7 11 14 12 10 8 6 4
%e 11 3 15 7 1 5 9 13 14 12 10 8 6 4
%e 11 3 15 7 1 5 9 13 16 14 12 10 8 6 4
%e 1 17 9 13 5 3 7 11 15 16 14 12 10 8 6 4
%e 1 17 9 13 5 3 7 11 15 18 16 14 12 10 8 6 4
%e 3 19 11 15 7 1 5 9 13 17 18 16 14 12 10 8 6 4
%e 3 19 11 15 7 1 5 9 13 17 20 18 16 14 12 10 8 6 4
%p sigr := proc(n,r)
%p floor(n/2^r) ;
%p end proc:
%p # A063695
%p f := proc(n)
%p local ndigs,fn,k ;
%p ndigs := convert(n,base,2) ;
%p fn := 0 ;
%p for k from 2 to nops(ndigs) by 2 do
%p fn := fn+op(k,ndigs)*2^(k-1)
%p end do;
%p fn ;
%p end proc:
%p g := proc(t,n)
%p local r;
%p if t =1 then
%p 0 ;
%p elif t > 1 then
%p r := ilog2( (n-1)/(t-1) ) ;
%p (-2)^r*(f( sigr(2*n-1,r) )+f( sigr(n-1,r) )-2*t+3) ;
%p end if;
%p end proc:
%p ft := proc(t,n)
%p f(n-1)+1+g(t,n) ;
%p end proc:
%p for n from 1 to 20 do
%p for t from 1 to n-1 do
%p printf("%3d ", ft(t,n)) ;
%p end do:
%p printf("\n") ;
%p end do:
%Y Cf. A090569 (column k=1).
%K nonn,tabl
%O 1,2
%A _R. J. Mathar_, Jun 22 2020
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