%I A147672
%S A147672 0,1,2,7,15,28,52,97,185,358,702,1387,2755,5488,10952,21877,43725,87418,
%T A147672 174802,349567,699095,1398148,2796252,5592457,11184865,22369678,
%U A147672 44739302,89478547,178957035,357914008,715827952
%N A147672 a(n)=a(n-2)+2^(n-1)+5 for n>3, a(0..3)=(0,1,2,7).
%C A147672 BRIDGE transform of A000079(n)=2^n. The BRIDGE transform of an increasing 
               sequence {c(n), n=0,1,2...} is the sequence b() defined as follows: 
               b(0)=0, b(1)=c(0), b(2)=c(1), b(3)=c(0)+c(1)+c(2) and for n>3, b(n)=b(n-2)+c(n-1)+2*c(1)+c(0).
%C A147672 The name comes from the puzzle "Crossing the bridge" (cf. link): If this 
               puzzle is generalized to a group of n people who need c(0),...,c(n-1) 
               minutes to cross the bridge, then b(n) is an upper bound for the 
               optimal solution, conjectured to be optimal, cf. explanations from 
               F. T. Adams-Watters to the SeqFan list, Nov 10 2008.
%C A147672 The BRIDGE transform can be generalized to a nonincreasing sequence as 
               given as PARI code. (Notice the vecsort() instruction.)
%H A147672 National Science Teachers Association, <a href="http://www.nsta.org/quantum/
               bridgarc.asp">Quantum CyberTeaser Archive #B205, May/June 1997</a>
%F A147672 2^(n-1) <= a(n) < 2^n for n>0.
%F A147672 G.f.: x(1-x+2x^2-x^3-6x^4)/((1+x)(1-2x)(1-x)^2). a(n) = 5n/2 -23/4 +(-1)^n/
               12 +2^(n+1)/3, n>1. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 10 2008]
%e A147672 a(4)=15=2+1+8+2+2 is the time required to cross the bridge for a boy, 
               his sister, his father and his mother if they require 1,2,4,8 minutes, 
               respectively, to cross the bridge individually (using the moves B+G,
               B,M+F,G,B+G).
%o A147672 (PARI) BRIDGE( a )={ local( s=vector(#a),t ); vector( #a, n, t=vecsort( 
               vecextract( a, 2^n-1 )); t[n]+if( n>3, t[1]+2*t[2]+BRIDGE( vecextract( 
               t,2^(n-2)-1 ))[n-2], if(n==3, t[1]+t[2] ))) }
%o A147672 A147672 = BRIDGE( vector(30,n,2^(n-1)) )
%Y A147672 Cf. A147673.
%Y A147672 Sequence in context: A061802 A003452 A000148 this_sequence A095091 A131412 
               A151998
%Y A147672 Adjacent sequences: A147669 A147670 A147671 this_sequence A147673 A147674 
               A147675
%K A147672 nonn
%O A147672 0,3
%A A147672 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 10 2008