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A075268 Trajectory of 442 under the Reverse and Add! operation carried out in base 2.

%I A075268 #14 Sep 08 2022 08:45:07
%S A075268 442,629,1326,2259,5508,6585,11628,15129,24912,26259,52038,77337,
%T A075268 155394,221931,442374,639009,1179738,1917027,3539130,5062869,10666542,
%U A075268 18285939,45369156,54513657,96444396,125792217,207562704,220034931
%N A075268 Trajectory of 442 under the Reverse and Add! operation carried out in base 2.
%C A075268 22, 77 and 442 are the first terms of A075252. The base 2 trajectory of 442 is completely different from the trajectories of 22 (cf. A061561) and 77 (cf. A075253). Using the formula given below one can prove that it does not contain a palindrome.
%C A075268 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
%C A075268 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
%C A075268 Interleaving of 2*A177420, A177421, 6*A177422, 3*A177423.
%H A075268 Reinhard Zumkeller, <a href="/A075268/b075268.txt">Table of n, a(n) for n = 0..1000</a>
%H A075268 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A075268 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A075268 a(0), ..., a(28) as above; a(29) = 703932681; a(30) =1310348526; a(31) = 2309980455; a(32) = 6143702712; a(33) = 7131271077; a(34) = 12699398352; a(35) = 13441412493; for n > 35 and
%F A075268 n = 0 (mod 4): a(n) = 3*2^(2*k+23)-12576771*2^k where k = (n-16)/4;
%F A075268 n = 1 (mod 4): a(n) = 3*2^(2*k+23)+12576771*2^k-3 where k = (n-17)/4;
%F A075268 n = 2 (mod 4): a(n) = 6*2^(2*k+23)-12576771*2^k where k = (n-18)/4;
%F A075268 n = 3 (mod 4): a(n) = 6*2^(2*k+23)+37730313*2^k-3 where k = (n-19)/4.
%F A075268 G.f.: (442+629*x+372*x^3+1530*x^4-192*x^5-2244*x^6-852*x^7-3784*x^8-8090*x^9 +5046*x^10+29034*x^11+47016*x^12+54354*x^13+79152*x^14+70254*x^15+65196*x^16 +358986*x^17+724128*x^18+334026*x^19+2081820*x^20+6043662*x^21+18678462*x^22+8601966*x^23 -23147244*x^24-15039648*x^25 -31927752*x^26-67877562*x^27+43880046*x^28+297766074*x^29 +396480108*x^30+734881086*x^31+3072255774*x^32+1018370430*x^33-3939844260*x^34-4608944376*x^35 -6616834356*x^36-3107825028*x^37+6655931736*x^38+7777900872*x^39+484428384*x^40 -2233413600*x^41-62899200*x^42+188697600*x^43) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
%F A075268 G.f. for the sequence starting at a(36): 3*x^36*(8455782368+8724086815*x -8321630144*x^2-8589934590*x^3-17045716960*x^4-18118934750*x^5+16911564736*x^6 +17984782524*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
%F A075268 a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%e A075268 442 (decimal) = 110111010 -> 110111010 + 010111011 = 1001110101 = 629 (decimal).
%t A075268 NestWhileList[# + IntegerReverse[#, 2] &, 442,  # !=
%t A075268 IntegerReverse[#, 2] &, 1, 27] (* _Robert Price_, Oct 18 2019 *)
%o A075268 (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 A075268 trajectory(442,28);
%o A075268 (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(442, 28, 2);
%o A075268 (Haskell)
%o A075268 a075268 n = a075268_list !! n
%o A075268 a075268_list = iterate a055944 442  -- _Reinhard Zumkeller_, Apr 21 2013
%Y A075268 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), 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 A075268 Cf. A177420 (a(4*n)/2), A177421 (a(4*n+1)), A177422 (a(4*n+2)/6), A177423 (a(4*n+3)/3).
%K A075268 base,nonn
%O A075268 0,1
%A A075268 _Klaus Brockhaus_, Sep 11 2002
%E A075268 Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by _Klaus Brockhaus_, May 07 2010