%I A051049
%S A051049 1,1,4,7,16,31,64,127,256,511,1024,2047,4096,8191,16384,32767,65536,
%T A051049 131071,262144,524287,1048576,2097151,4194304,8388607,16777216,
%U A051049 33554431,67108864,134217727,268435456,536870911,1073741824
%N A051049 Number of moves needed to solve an n-ring baguenaudier if the two end 
               rings can be moved simultaneously.
%C A051049 Sum of terms of rows of triangle from submitted A166692=1,0,1,1,1,2,. 
               Binomial transform of (A166265=1,3*A001045) signed. Main diagonal 
               and upper are 1,3,6,12,=A003945. a(n) - A001045(n+1) =0,A000975 (submitted, 
               last year ?). Differences:A062510,Jacobsthal. [From Paul Curtz (bpcrtz(AT)free.fr), 
               Oct 20 2009]
%H A051049 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Baguenaudier.html">Link to a section of The World of Mathematics.</
               a>
%F A051049 a(n)=(2^(n+1)-(1+(-1)^(n+1)))/2. - Paul Barry (pbarry(AT)wit.ie), Apr 
               24 2003
%F A051049 a(n+2)=a(n+1)+2a(n)+1, a(0)=a(1)=1 - Paul Barry (pbarry(AT)wit.ie), May 
               01 2003
%F A051049 a(n)= (sum{k=0..n, 2^k}+(-1)^n)/2=(A000225(n+1)+(-1)^n)/2. - Paul Barry 
               (pbarry(AT)wit.ie), May 27 2003
%F A051049 (a(n+1)-a(n))/3=A001045(n) - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A051049 a(n)=sum{k=0..floor(n/2), C(n+1, 2k) } - Paul Barry (pbarry(AT)wit.ie), 
               May 27 2003
%F A051049 a(n)=sum{k=0..n, C(n, k)+(-1)^(n-k)}-1 - Paul Barry (pbarry(AT)wit.ie), 
               Jul 21 2003
%F A051049 G.f.:(1-x+x^2)/((1-x^2)(1-2x)); E.g.f.: exp(2x)-sinh(x). - Paul Barry 
               (pbarry(AT)wit.ie), Sep 19 2003
%F A051049 a(n)=sum{k=0..n, sum{j=0..n-k, if(mod(j-k, 2)=0, binomial(n-k, j), 0}}; 
               - Paul Barry (pbarry(AT)wit.ie), Jan 25 2005
%F A051049 Row sums of triangle A135221 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 23 2007
%t A051049 b[n_?EvenQ] := 2^(n - 1) - 1; b[n_?OddQ] := 2^(n - 1); Table[b[n], {n, 
               50}]]
%Y A051049 Cf. A000975. Row sums of A131086.
%Y A051049 Cf. A135221.
%Y A051049 Sequence in context: A093210 A133600 A166865 this_sequence A108122 A027609 
               A145763
%Y A051049 Adjacent sequences: A051046 A051047 A051048 this_sequence A051050 A051051 
               A051052
%K A051049 nonn
%O A051049 0,3
%A A051049 Eric Weisstein (eric(AT)weisstein.com)