A075466 Trajectory of 266718 under the Reverse and Add! operation carried out in base 4, written in base 10.

266718, 1017375, 2019150, 4934715, 20413980, 34239885, 64220175, 127195950, 321080475, 1286586060, 2154739965, 4288508415, 8571775230, 21401016315, 85781907180, 149736661725, 278082371775, 1369020907200, 1433193762225

Offset: 0

Author

Klaus Brockhaus, Sep 18 2002

Keywords

• base
• nonn

Comments

266718 = A075421(358) is the smallest term > 318 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(26) onward; the g.f. for the complete sequence is known but more than twice as big.

Examples

			266718 (decimal) = 1001013132 -> 1001013132 + 2313101001 = 3320120133 = 1017375 (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, A075467.

Programs

Mathematica

NestWhileList[# + IntegerReverse[#, 4] &, 266718,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

PARI

{m=266718; 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(18) as above; a(19) = 2780823717750; a(20) = 5492189757120; a(21) = 5749636151985; a(22) = 11156010444150; a(23) = 21968759028480; a(24) = 22226205423345; a(25) = 44109148986870; for n > 25 and n = 2 (mod 6): a(n) = 5*4^(2*k+14)-83865605*4^k where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+14)[phone]*4^k-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+14)[phone]*4^k-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+14)-335462420*4^k where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+14)[phone]*4^k-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+14)[phone]*4^k-10 where k = (n-7)/6. G.f.: -15*(47049901525664*x^11+23708157972464*x^10+23433347158016*x^9-46912496118440*x^8-23502049861628*x^7-23433347158016*x^6-11908468626600*x^5-6137441522940*x^4-5862630708480*x^3+11771063219370*x^2+5931333412095*x+5862630708480)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))