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A077077
Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.
9
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.
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
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