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A075467
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Trajectory of 270798 under the Reverse and Add! operation carried out in base 4, written in base 10.
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6
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270798, 1005135, 1994670, 5058075, 20047500, 33313725, 66545850, 112201785, 225464610, 368353785, 835135950, 1587633825, 2841028950, 5347819200, 5598498225, 10862757750, 21453946560, 22456662705, 43576370550
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OFFSET
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0,1
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COMMENTS
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The base 4 trajectory of 270798 = A075421(370) 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. - The generating function given describes the sequence from a(11) onward; the g.f. for the complete sequence is known but nearly twice as big.
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LINKS
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FORMULA
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a(0), ..., a(10) as above; for n > 10 and n = 5 (mod 6): a(n) = 5*4^(2*k+10)+15341035*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+10)+9792150*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 20*4^(2*k+10)-1305620*4^k where k = (n-1)/6; n = 2 (mod 6): a(n) = 20*4^(2*k+10)+14361820*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+10)+7833720*4^k-10 where k = (n-3)/6; n = 4 (mod 6): a(n) = 80*4^(2*k+10)-1305620*4^k where k = (n-4)/6. G.f.: -15*(1426085120*x^11+749251744*x^10+419191024*x^9-1430263104*x^8-715827880*x^7-369055228*x^6-352343296*x^5-222825800*x^4-155978060*x^3+356521280*x^2+189401930*x+105842255)/((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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270798 (decimal) = 1002013032 -> 1002013032 + 2303102001 = 3311121033 = 1005135 (decimal).
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MATHEMATICA
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NestWhileList[# + IntegerReverse[#, 4] &, 270798, # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)
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PROG
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(PARI) {m=270798; stop=20; 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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