|
%I #14 Sep 08 2022 08:45:07
%S 775,1674,2325,5022,8919,23976,26757,47376,49581,96048,102669,193056,
%T 197469,388704,401949,779328,788157,1563840,1590333,3131520,3149181,
%U 6273408,6326397,12554496,12589821,25129728,25235709,50274816,50345469
%N Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.
%C The base 2 trajectory of 775 = A075252(5) provably does not contain a palindrome. A proof can be based on the formula given below.
%C lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.
%C lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
%C Interleaving of A177843, 6*A177844, 3*A177845, 6*A177846.
%H Reinhard Zumkeller, <a href="/A077077/b077077.txt">Table of n, a(n) for n = 0..1000</a>
%H Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F a(0), ..., a(5) as above; for n > 5 and
%F n = 2 (mod 4): a(n) = 3*2^(2*k+7)+273*2^k-3 where k = (n+6)/4;
%F n = 3 (mod 4): a(n) = 6*2^(2*k+7)-222*2^k where k = (n+5)/4;
%F n = 0 (mod 4): a(n) = 6*2^(2*k+7)+54*2^k-3 where k = (n+4)/4;
%F n = 1 (mod 4): a(n) = 12*2^(2*k+7)-282*2^k where k = (n+3)/4.
%F a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)+2*a(n-4)+2*a(n-5)-4*a(n-6)-4*a(n-7)-3 for n > 12; a(0), ..., a(12) as above.
%F G.f.: (775+1674*x+1944*x^4+8910*x^5+4650*x^6-14508*x^7-19840*x^8-22644*x^9- 1860*x^10+28680*x^11+14328*x^12-2112*x^13) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
%F G.f. for the sequence starting at a(6): 3*(8919+15792*x-10230*x^2- 15360*x^3-15358*x^4-31696*x^5+16668*x^6+31264*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
%F a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%e 775 (decimal) = 1100000111 -> 1100000111 + 1110000011 = 11010001010 = 1674 (decimal).
%t NestWhileList[# + IntegerReverse[#, 2] &, 775, # !=
%t IntegerReverse[#, 2] &, 1, 28] (* _Robert Price_, Oct 18 2019 *)
%o (PARI) trajectory(n,steps) = {local(v, k=n); for(j=0, steps, print1(k, ", "); v=binary(k); k+=sum(j=1, #v, 2^(j-1)*v[j]))};
%o trajectory(775,28);
%o (Magma) trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:= Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(775, 28, 2);
%o (Haskell)
%o a077077 n = a077077_list !! n
%o a077077_list = iterate a055944 775 -- _Reinhard Zumkeller_, Apr 21 2013
%Y Cf. A058042 (trajectory of 22 in base 2, written in base 2), A061561 (trajectory of 22 in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n).
%Y Cf. A177843 (a(4*n)), A177844 (a(4*n+1)/6), A177845 (a(4*n+2)/3), A177846 (a(4*n+3)/6).
%K base,nonn
%O 0,1
%A _Klaus Brockhaus_, Oct 25 2002
%E Comment edited, three comments and formula added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by _Klaus Brockhaus_, May 14 2010
| 