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

Author

Klaus Brockhaus, Sep 11 2002

Keywords

• base
• nonn

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.

Examples

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

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!

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).

Programs

Haskell

a075268 n = a075268_list !! n
a075268_list = iterate a055944 442  -- Reinhard Zumkeller, Apr 21 2013

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);

Mathematica

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

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);

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)[phone]*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)[phone]*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 [phone]*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 [phone]*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

Extensions

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