A076247 Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

1059774, 4187583, 8355006, 20822715, 83391660, 144328605, 268919295, 1339676160, 1349598705, 2683144950, 5361370860, 9358549725, 17380163775, 85563883200, 89574690225, 173801637750, 343262166720, 359352580785

Offset: 0

Author

Klaus Brockhaus, Oct 03 2002

Keywords

• base
• nonn

Comments

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.

Examples

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

Links

• Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
• Index entries for sequences related to Reverse and Add!

Crossrefs

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

Programs

Mathematica

NestWhileList[# + IntegerReverse[#, 4] &, 1059774,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

PARI

{m=1059774; stop=19; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

Formula

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)[phone]*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+12)[phone]*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)[phone]*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+12)[phone]*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 [phone]*x+22884144448)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))