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A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

%I A075153 #14 Sep 08 2022 08:45:07
%S A075153 318,1071,5040,5985,10710,20400,24225,43350,81600,85425,165750,327360,
%T A075153 342705,664950,1309440,1324785,2629110,5241600,5303025,10524150,
%U A075153 20966400,21027825,41973750,83880960,84126705,167925750,335523840
%N A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.
%C A075153 290 is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome. 318 (not 255 since 255 is a base 4 palindrome) is up to now the smallest number whose base 4 trajectory provably does not contain a palindrome. 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 A075153 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 3 in {1, 2}.
%C A075153 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.
%H A075153 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A075153 K. S. Brown, <a href="http://www.mathpages.com/home/kmath004/kmath004.htm">Digit Reversal Sums Leading to Palindromes</a>
%H A075153 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A075153 a(0) = 318; a(1) = 1071; for n > 1 and n = 2 (mod 6): a(n) = 5*4^(2*k+5)-5*4^(k+2) where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+5)+55*4^(k+2)-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+5)+30*4^(k+2)-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+5)-5*4^(k+2) where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+5)+235*4^(k+2)-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+5)+150*4^(k+2)-10 where k = (n-7)/6.
%F A075153 G.f.: 3*(106 +357*x +1680*x^2 +1465*x^3 +1785*x^4 -1600*x^5 -1900*x^6 -3400*x^7 -6800*x^8 -9780*x^9 -9860*x^10 +6720*x^11 +10064*x^12 +11088*x^13) / ((1-x)*(1+x+x^2)*(1-2*x^3)*(1+2*x^3)*(1-4*x^3)).
%e A075153 318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).
%t A075153 NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
%t A075153 IntegerReverse[#, 4] &, 1, 26] (* _Robert Price_, Oct 18 2019 *)
%o A075153 (PARI) {m=318; stop=29; c=0; while(c<stop,print1(k=m,","); rev=0; while(k>0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}
%o A075153 (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(318, 26, 4);
%Y A075153 Cf. A058042 (trajectory of binary number 10110 (decimal 22)), A061561 (A058042 written in base 10), A066450 (conjectured minimal k so that the trajectory of k in base n does not lead to a palindrome).
%Y A075153 Cf. A075253 (trajectory of 77 in base 2), A075420 (trajectory of n in base 4 (presumably) does not reach a palindrome), A075421 (trajectory of n in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075299 (trajectory of 290 in base 4), A075466 (trajectory of 266718 in base 4), A075467 (trajectory of 270798 in base 4), A076247 (trajectory of 1059774 in base 4), A076248 (trajectory of 1059831 in base 4), A091675 (trajectory of n in base 4 (presumably) does not join the trajectory of any m < n).
%Y A075153 Cf. A166912 (a(6*n)/3), A166913 (a(6*n+1)/3), A166914 (a(6*n+2)/240), A166915 (a(6*n+3)/15), A166916 (a(6*n+4)/30), A166917 (a(6*n+5)/240).
%K A075153 base,nonn
%O A075153 0,1
%A A075153 _Klaus Brockhaus_, Sep 05 2002
%E A075153 Two comments added, g.f. edited, MAGMA program and cross-references added by _Klaus Brockhaus_, Oct 26 2009