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A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.

%I A075253 #29 Jun 21 2023 09:12:12
%S A075253 77,166,267,684,897,1416,1557,2904,3333,5904,6189,11952,12813,24096,
%T A075253 24669,48480,50205,97344,98493,195264,198717,391296,393597,783744,
%U A075253 790653,1569024,1573629,3140352,3154173,6283776,6292989,12572160
%N A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.
%C A075253 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.
%C A075253 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
%C A075253 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
%C A075253 Interleaving of A176632, 2*A176633, 3*A176634, 12*A176635.
%C A075253 From _A.H.M. Smeets_, Feb 11 2019: (Start)
%C A075253 Pattern with cycle length 4 in binary representation, represented by contextfree grammars with production rules:
%C A075253 S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1100010;
%C A075253 S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 0000101;
%C A075253 S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101011;
%C A075253 S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0100000;
%C A075253 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)
%H A075253 Reinhard Zumkeller, <a href="/A075253/b075253.txt">Table of n, a(n) for n = 0..1000</a>
%H A075253 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A075253 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%H A075253 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 3, 0, 0, 0, -6, 0, 4).
%F A075253 a(0) = 77; a(1) = 166; a(2) = 267; for n > 2 and
%F A075253 n = 3 (mod 4): a(n) = 48*2^(2*k)-21*2^k where k = (n+5)/4;
%F A075253 n = 0 (mod 4): a(n) = 48*2^(2*k)+33*2^k-3 where k = (n+4)/4;
%F A075253 n = 1 (mod 4): a(n) = 96*2^(2*k)-30*2^k where k = (n+3)/4;
%F A075253 n = 2 (mod 4): a(n) = 96*2^(2*k)+6*2^k-3 where k = (n+2)/4.
%F A075253 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)).
%F A075253 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)).
%F A075253 a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%e A075253 267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).
%p A075253 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
%t A075253 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 *)
%t A075253 NestWhileList[# + IntegerReverse[#, 2] &, 77,  # !=
%t A075253 IntegerReverse[#, 2] &, 1, 31] (* _Robert Price_, Oct 18 2019 *)
%o A075253 (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)}
%o A075253 (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);
%o A075253 (Haskell)
%o A075253 a075253 n = a075253_list !! n
%o A075253 a075253_list = iterate a055944 77  -- _Reinhard Zumkeller_, Apr 21 2013
%o A075253 (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
%Y A075253 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).
%Y A075253 Cf. A176632 (a(4*n)), A176633 (a(4*n+1)/2), A176634 (a(4*n+2)/3), A176635 (a(4*n+3)/12).
%K A075253 base,nonn
%O A075253 0,1
%A A075253 _Klaus Brockhaus_, Sep 10 2002
%E A075253 Three comments added, g.f. edited, MAGMA program and crossrefs added by _Klaus Brockhaus_, Apr 25 2010