A075467 Trajectory of 270798 under the Reverse and Add! operation carried out in base 4, written in base 10.

270798, 1005135, 1994670, 5058075, 20047500, 33313725, 66545850, 112201785, 225464610, 368353785, 835135950, 1587633825, 2841028950, 5347819200, 5598498225, 10862757750, 21453946560, 22456662705, 43576370550

Offset: 0

Author

Klaus Brockhaus, Sep 18 2002

Keywords

• base
• nonn

Comments

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.

Examples

			270798 (decimal) = 1002013032 -> 1002013032 + 2303102001 = 3311121033 = 1005135 (decimal).
		

Links

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

Crossrefs

Cf. A075153, A075421, A075466.

Programs

Mathematica

NestWhileList[# + IntegerReverse[#, 4] &, 270798,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

PARI

{m=270798; stop=20; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

Formula

a(0), ..., a(10) as above; for n > 10 and n = 5 (mod 6): a(n) = 5*4^(2*k+10)[phone]*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)[phone]*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))