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A076248
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Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.
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2
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1059831, 4728312, 7831065, 14433270, 24913965, 56412450, 92165625, 208908750, 396926625, 710289750, 1336954560, 1398889905, 2715199350, 5363547840, 5614238385, 10894222710, 21453945600, 21701687025, 43073052150
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OFFSET
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0,1
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COMMENTS
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1059831 = A075421(1105 ) is the fifth term of A075421 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.
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LINKS
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FORMULA
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a(0), ..., a(7) as above; for n > 7 and n = 2 (mod 6): a(n) = 5*4^(2*k+9)+3836395*4^k-15 where k = (n+4)/6; n = 3 (mod 6): a(n) = 10*4^(2*k+9)+2450070*4^k-10 where k = (n+3)/6; n = 4 (mod 6): a(n) = 20*4^(2*k+9)-326420*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+9)+3544540*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 40*4^(2*k+9)+1927800*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 80*4^(2*k+9)-322580*4^k where k = (n-1)/6. G.f.: -3*(668508000*x^19+444361200*x^18+222142800*x^17-528080680*x^16-356464620*x^15 -125753060*x^14-299532884*x^13-188180432*x^12-143040640*x^11+128992350*x^10+90219415*x^9 +38288125*x^8+28112975*x^7+6666425*x^6+5752375*x^5+424135*x^4+3044705*x^3+2610355*x^2 + 1576104*x+353277)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))
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EXAMPLE
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1059831 (decimal) = 10002233313 -> 10002233313 + 31333220001 = 102002113320 = 4728312 (decimal).
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MATHEMATICA
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NestWhileList[# + IntegerReverse[#, 4] &, 1059831, # != IntegerReverse[ #, 4] &, 1, 23] (* Robert Price, Oct 19 2019 *)
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PROG
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(PARI) {m=1059831; stop=19; 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)}
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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