login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2. 13
77, 166, 267, 684, 897, 1416, 1557, 2904, 3333, 5904, 6189, 11952, 12813, 24096, 24669, 48480, 50205, 97344, 98493, 195264, 198717, 391296, 393597, 783744, 790653, 1569024, 1573629, 3140352, 3154173, 6283776, 6292989, 12572160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

22 is the smallest number whose base 2 trajectory (A061561) provably does not contain a palindrome. 77 is the next number (cf. A075252) with a completely different trajectory which has this property. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.

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 A176632, 2*A176633, 3*A176634, 12*A176635.

From A.H.M. Smeets, Feb 11 2019: (Start)

Pattern with cycle length 4 in binary representation, represented by contextfree grammars with production rules:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1100010;

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 0000101;

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101011;

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0100000;

the trajectory is similar to that of 22 (see A058042) except for the stopping strings in T_a, T_b, T_c and T_d. (End)

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) = 77; a(1) = 166; a(2) = 267; for n > 2 and

n = 3 (mod 4): a(n) = 48*2^(2*k)-21*2^k where k = (n+5)/4;

n = 0 (mod 4): a(n) = 48*2^(2*k)+33*2^k-3 where k = (n+4)/4;

n = 1 (mod 4): a(n) = 96*2^(2*k)-30*2^k where k = (n+3)/4;

n = 2 (mod 4): a(n) = 96*2^(2*k)+6*2^k-3 where k = (n+2)/4.

G.f.: (77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8 +632*x^9+504*x^10) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).

G.f. for the sequence starting at a(3): 3*x^3*(228+299*x-212*x^2 -378*x^3-448*x^4-446*x^5+432*x^6+524*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

267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).

MAPLE

seq(coeff(series((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8+632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Feb 12 2019

MATHEMATICA

CoefficientList[Series[(77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)), {x, 0, 40}], x] (* G. C. Greubel, Feb 11 2019 *)

NestWhileList[# + IntegerReverse[#, 2] &, 77,  # !=

IntegerReverse[#, 2] &, 1, 31] (* Robert Price, Oct 18 2019 *)

PROG

(PARI) {m=77; stop=34; c=0; while(c<stop, print1(k=m, ", "); rev=0; while(k>0, d=divrem(k, 2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}

(MAGMA) trajectory:=function(init, steps, base) S:=[init]; a:=S[1]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a, base)), base); Append(~S, a); end for; return S; end function; trajectory(77, 31, 2);

(Haskell)

a075253 n = a075253_list !! n

a075253_list = iterate a055944 77  -- Reinhard Zumkeller, Apr 21 2013

(Sage) ((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 11 2019

CROSSREFS

Cf. A061561 (trajectory of 22 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 presumably does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and presumably does not join the trajectory of any term m < n), A092210 (trajectory of n in base 2 presumably does not join the trajectory of any m < n).

Cf. A176632 (a(4*n)), A176633 (a(4*n+1)/2), A176634 (a(4*n+2)/3), A176635 (a(4*n+3)/12).

Sequence in context: A044709 A338189 A308963 * A217790 A046513 A199994

Adjacent sequences:  A075250 A075251 A075252 * A075254 A075255 A075256

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Sep 10 2002

EXTENSIONS

Three comments added, g.f. edited, MAGMA program and crossrefs added by Klaus Brockhaus, Apr 25 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)