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A077077 Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.

%I A077077 #14 Sep 08 2022 08:45:07
%S A077077 775,1674,2325,5022,8919,23976,26757,47376,49581,96048,102669,193056,
%T A077077 197469,388704,401949,779328,788157,1563840,1590333,3131520,3149181,
%U A077077 6273408,6326397,12554496,12589821,25129728,25235709,50274816,50345469
%N A077077 Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.
%C A077077 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 A077077 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.
%C A077077 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
%C A077077 Interleaving of A177843, 6*A177844, 3*A177845, 6*A177846.
%H A077077 Reinhard Zumkeller, <a href="/A077077/b077077.txt">Table of n, a(n) for n = 0..1000</a>
%H A077077 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A077077 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A077077 a(0), ..., a(5) as above; for n > 5 and
%F A077077 n = 2 (mod 4): a(n) = 3*2^(2*k+7)+273*2^k-3 where k = (n+6)/4;
%F A077077 n = 3 (mod 4): a(n) = 6*2^(2*k+7)-222*2^k where k = (n+5)/4;
%F A077077 n = 0 (mod 4): a(n) = 6*2^(2*k+7)+54*2^k-3 where k = (n+4)/4;
%F A077077 n = 1 (mod 4): a(n) = 12*2^(2*k+7)-282*2^k where k = (n+3)/4.
%F A077077 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 A077077 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 A077077 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 A077077 a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%e A077077 775 (decimal) = 1100000111 -> 1100000111 + 1110000011 = 11010001010 = 1674 (decimal).
%t A077077 NestWhileList[# + IntegerReverse[#, 2] &, 775,  # !=
%t A077077 IntegerReverse[#, 2] &, 1, 28] (* _Robert Price_, Oct 18 2019 *)
%o A077077 (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 A077077 trajectory(775,28);
%o A077077 (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 A077077 (Haskell)
%o A077077 a077077 n = a077077_list !! n
%o A077077 a077077_list = iterate a055944 775  -- _Reinhard Zumkeller_, Apr 21 2013
%Y A077077 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 A077077 Cf. A177843 (a(4*n)), A177844 (a(4*n+1)/6), A177845 (a(4*n+2)/3), A177846 (a(4*n+3)/6).
%K A077077 base,nonn
%O A077077 0,1
%A A077077 _Klaus Brockhaus_, Oct 25 2002
%E A077077 Comment edited, three comments and formula added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by _Klaus Brockhaus_, May 14 2010