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A075153
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Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.
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15
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318, 1071, 5040, 5985, 10710, 20400, 24225, 43350, 81600, 85425, 165750, 327360, 342705, 664950, 1309440, 1324785, 2629110, 5241600, 5303025, 10524150, 20966400, 21027825, 41973750, 83880960, 84126705, 167925750, 335523840
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OFFSET
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0,1
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COMMENTS
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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.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 3 in {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.
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LINKS
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FORMULA
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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.
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)).
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EXAMPLE
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318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).
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MATHEMATICA
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NestWhileList[# + IntegerReverse[#, 4] &, 318, # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)
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PROG
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(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)}
(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);
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CROSSREFS
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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).
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).
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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Two comments added, g.f. edited, MAGMA program and cross-references added by Klaus Brockhaus, Oct 26 2009
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STATUS
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approved
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