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A075268
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Trajectory of 442 under the Reverse and Add! operation carried out in base 2.
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11
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442, 629, 1326, 2259, 5508, 6585, 11628, 15129, 24912, 26259, 52038, 77337, 155394, 221931, 442374, 639009, 1179738, 1917027, 3539130, 5062869, 10666542, 18285939, 45369156, 54513657, 96444396, 125792217, 207562704, 220034931
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OFFSET
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0,1
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COMMENTS
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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.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
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LINKS
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FORMULA
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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
n = 0 (mod 4): a(n) = 3*2^(2*k+23)-12576771*2^k where k = (n-16)/4;
n = 1 (mod 4): a(n) = 3*2^(2*k+23)+12576771*2^k-3 where k = (n-17)/4;
n = 2 (mod 4): a(n) = 6*2^(2*k+23)-12576771*2^k where k = (n-18)/4;
n = 3 (mod 4): a(n) = 6*2^(2*k+23)+37730313*2^k-3 where k = (n-19)/4.
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)).
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)).
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EXAMPLE
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442 (decimal) = 110111010 -> 110111010 + 010111011 = 1001110101 = 629 (decimal).
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MATHEMATICA
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NestWhileList[# + IntegerReverse[#, 2] &, 442, # !=
IntegerReverse[#, 2] &, 1, 27] (* Robert Price, Oct 18 2019 *)
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PROG
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(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]))};
trajectory(442, 28);
(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);
(Haskell)
a075268 n = a075268_list !! n
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CROSSREFS
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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).
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 07 2010
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STATUS
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approved
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