A077077 Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.

775, 1674, 2325, 5022, 8919, 23976, 26757, 47376, 49581, 96048, 102669, 193056, 197469, 388704, 401949, 779328, 788157, 1563840, 1590333, 3131520, 3149181, 6273408, 6326397, 12554496, 12589821, 25129728, 25235709, 50274816, 50345469

Offset

0,1

Comments

The base 2 trajectory of 775 = A075252(5) 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 A177843, 6*A177844, 3*A177845, 6*A177846.

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(5) as above; for n > 5 and

n = 2 (mod 4): a(n) = 3*2^(2*k+7)+273*2^k-3 where k = (n+6)/4;

n = 3 (mod 4): a(n) = 6*2^(2*k+7)-222*2^k where k = (n+5)/4;

n = 0 (mod 4): a(n) = 6*2^(2*k+7)+54*2^k-3 where k = (n+4)/4;

n = 1 (mod 4): a(n) = 12*2^(2*k+7)-282*2^k where k = (n+3)/4.

a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)+2*a(n-4)+2*a(n-5)-4*a(n-6)-4*a(n-7)-3 for n > 12; a(0), ..., a(12) as above.

G.f.: (775+1674*x+1944*x^4+8910*x^5+4650*x^6-14508*x^7-19840*x^8-22644*x^9- 1860*x^10+28680*x^11+14328*x^12-2112*x^13) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).

G.f. for the sequence starting at a(6): 3*(8919+15792*x-10230*x^2- 15360*x^3-15358*x^4-31696*x^5+16668*x^6+31264*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

775 (decimal) = 1100000111 -> 1100000111 + 1110000011 = 11010001010 = 1674 (decimal).

Mathematica

NestWhileList[# + IntegerReverse[#, 2] &, 775,  # !=
IntegerReverse[#, 2] &, 1, 28] (* 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(775, 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(775, 28, 2);
(Haskell)
a077077 n = a077077_list !! n
a077077_list = iterate a055944 775  -- 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), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 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. A177843 (a(4*n)), A177844 (a(4*n+1)/6), A177845 (a(4*n+2)/3), A177846 (a(4*n+3)/6).

Sequence in context: A233991 A250087 A252673 * A177845 A177843 A202893

Adjacent sequences: A077074 A077075 A077076 * A077078 A077079 A077080

Keyword

base,nonn

Author

Klaus Brockhaus, Oct 25 2002

Extensions

Comment edited, three comments and formula added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 14 2010

Status

approved