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A077076
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Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.
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9
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537, 1146, 1899, 3618, 4713, 9522, 14427, 28386, 37533, 84966, 138123, 353004, 466209, 738024, 833301, 1525224, 1718853, 3048912, 3239469, 6196176, 6583437, 12389280, 12770397, 24975264, 25749789, 49944384, 50706621, 100282176
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OFFSET
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0,1
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COMMENTS
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The base 2 trajectory of 537 = A075252(4) provably does not contain a palindrome. A proof can be based on the formula given below.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
Interleaving of 3*A177682, 6*A177683, 3*A177684, 6*A177685.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!
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FORMULA
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a(0), ..., a(11) as above; for n > 11 and
n = 0 (mod 4): a(n) = 3*2^(2*k+13)+18249*2^k-3 where k = (n-4)/4;
n = 1 (mod 4): a(n) = 6*2^(2*k+13)-12102*2^k where k = (n-5)/4;
n = 2 (mod 4): a(n) = 6*2^(2*k+13)+11718*2^k-3 where k = (n-6)/4;
n = 3 (mod 4): a(n) = 12*2^(2*k+13)-11910*2^k where k = (n-7)/4.
G.f.: 3*(179+382*x+96*x^2+60*x^3-328*x^4-444*x^5+1170*x^6+2232*x^7 +1166*x^8+5644*x^9+15402*x^10+46922*x^11+39850*x^12-62920*x^13-132612*x^14 -97532*x^15-34148*x^16+83800*x^17+109224*x^18+21856*x^19) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
G.f. for the sequence starting at a(12): 3*x^12*(155403+246008*x-188442*x^2-229616*x^3-260350*x^4-508920*x^5+293388*x^6+492528*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4))
a(n+1) = A055944(a(n)). - Reinhard Zumkeller, Apr 21 2013
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EXAMPLE
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537 (decimal) = 1000011001 -> 1000011001 + 1001100001 = 10001111010= 1146 (decimal).
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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(537, 27);
(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(537, 27, 2);
(Haskell)
a077076 n = a077076_list !! n
a077076_list = iterate a055944 537 -- Reinhard Zumkeller, Apr 21 2013
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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), A075268 (trajectory of 442 in base 2), A077077 (trajectory of 775 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).
Cf. A177682 (a(4*n)/3), A177683 (a(4*n+1)/6), A177684 (a(4*n+2)/3), A177685 (a(4*n+3)/6).
Sequence in context: A183598 A067723 A059949 * A033916 A206067 A183725
Adjacent sequences: A077073 A077074 A077075 * A077077 A077078 A077079
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KEYWORD
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base,nonn
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AUTHOR
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Klaus Brockhaus, Oct 25 2002
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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 12 2010
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STATUS
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approved
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