A076247 - OEIS

%I #10 Oct 18 2019 21:12:55

%S 1059774,4187583,8355006,20822715,83391660,144328605,268919295,

%T 1339676160,1349598705,2683144950,5361370860,9358549725,17380163775,

%U 85563883200,89574690225,173801637750,343262166720,359352580785

%N Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

%C 1059774 = A075421(1096) is the fourth 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. - The generating function given describes the sequence from a(16) onward; the g.f. for the complete sequence is known but nearly twice as big.

%H Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>

%H <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>

%F a(0), ..., a(15) as above; for n > 15 and n = 4 (mod 6): a(n) = 5*4^(2*k+12)-5237765*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 5*4^(2*k+12)+246174955*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+12)+157132950*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 20*4^(2*k+12)-20951060*4^k where k = (n-1)/6; n = 2 (mod 6): a(n) = 20*4^(2*k+12)+230461660*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+12)+125706360*4^k-10 where k = (n-3)/6. G.f.: -15*(185397326496*x^11+95559181296*x^10+91268404224*x^9-183251937960*x^8-92341098492*x^7 -91268404224*x^6-48628806952*x^5-27174921532*x^4-22884144448*x^3+46483418410*x^2 +23956838719*x+22884144448)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

%e 1059774 (decimal) = 10002232332 -> 10002232332 + 23323220001 = 33332112333 = 4187583 (decimal).

%t NestWhileList[# + IntegerReverse[#, 4] &, 1059774,  # !=

%t IntegerReverse[#, 4] &, 1, 23] (* _Robert Price_, Oct 18 2019 *)

%o (PARI) {m=1059774; 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)}

%Y Cf. A075421, A075153, A075466, A075467, A076248.

%K base,nonn

%O 0,1

%A _Klaus Brockhaus_, Oct 03 2002