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

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

Offset

0,1

Comments

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.

Interleaving of 2*A177420, A177421, 6*A177422, 3*A177423.

Links

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!

Formula

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)).

a(n+1) = A055944(a(n)). - Reinhard Zumkeller, Apr 21 2013

Example

442 (decimal) = 110111010 -> 110111010 + 010111011 = 1001110101 = 629 (decimal).

Mathematica

NestWhileList[# + IntegerReverse[#, 2] &, 442,  # !=
IntegerReverse[#, 2] &, 1, 27] (* Robert Price, Oct 18 2019 *)

Prog

(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
a075268_list = iterate a055944 442  -- Reinhard Zumkeller, Apr 21 2013

Crossrefs

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).

Cf. A177420 (a(4*n)/2), A177421 (a(4*n+1)), A177422 (a(4*n+2)/6), A177423 (a(4*n+3)/3).

Sequence in context: A110996 A013769 A013899 * A332531 A158322 A031720

Adjacent sequences: A075265 A075266 A075267 * A075269 A075270 A075271

Keyword

base,nonn

Author

Klaus Brockhaus, Sep 11 2002

Extensions

Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 07 2010

Status

approved