A092210 -id:A092210 - cp's OEIS frontend

A092212 a(n) = smallest non-palindromic k such that the base-2 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A092210(n).

26, 65649, 89, 4193, 3599, 775, 68076, 2173

Offset: 1

Klaus Brockhaus, Feb 25 2004

Terms a(9) to a(29) are 205796147 (conjectured), 4402, 16720, 1089448, 442, 537, unknown, 1050177, 1575, 28822, unknown, 40573, 1066, 1587, unknown, unknown, 1081, 1082, 1085, 1115, 4185.
a(n) >= A092210(n); a(n) = A092210(n) iff the trajectory of A092210(n) is palindrome-free, i.e., A092210(n) is also a term of A075252.
a(n) determines a 1-to-1 mapping from the terms of A092210 to the terms of A075252, the inverse of the mapping determined by A092211.
The 1-to-1 property of the mapping depends on the conjecture that the base-2 Reverse and Add! trajectory of each term of A092210 contains only a finite number of palindromes (cf. A092215).
Base-2 analog of A089494 (base 10) and A091677 (base 4).

			A092210(3) = 64, the trajectory of 64 joins the trajectory of 89 at 48480, so a(3) = 89. A092210(5) = 98, the trajectory of 98 joins the trajectory of 3599 = A075252(16) at 401104704, so a(5) = 3599.

Index entries for sequences related to Reverse and Add!

Cf. A075252, A092210, A092211, A092215, A089494, A091677.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = NestList[# + IntegerReverse[#, 2] &, 1, limit];
A092210 = Flatten@{1, Select[Range[2, 266], (l =
        Length@NestWhileList[# + IntegerReverse[#, 2] &, #, !
            MemberQ[utraj, #] &, 1, limit];
       utraj =
        Union[utraj, NestList[# + IntegerReverse[#, 2] &, #, limit]];
       l == limit + 1) &]};
A092212 = {};
For[i = 1, i <= Length@A092210, i++,
k = A092210[[i]];
itraj = NestList[# + IntegerReverse[#, 2] &, A092210[[i]], limit];
While[ktraj =
   NestWhileList[# + IntegerReverse[#, 2] &,
    k, # != IntegerReverse[#, 2] &, 1, limit];
  PalindromeQ[k] || Length@ktraj != limit + 1 || ! IntersectingQ[itraj, ktraj], k++];
AppendTo[A092212, k]]; A092212 (* Robert Price, Nov 03 2019 *)

a(1) and a(3) corrected by Robert Price, Nov 06 2019

A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.

Original entry on oeis.org

77, 166, 267, 684, 897, 1416, 1557, 2904, 3333, 5904, 6189, 11952, 12813, 24096, 24669, 48480, 50205, 97344, 98493, 195264, 198717, 391296, 393597, 783744, 790653, 1569024, 1573629, 3140352, 3154173, 6283776, 6292989, 12572160

Offset: 0

Klaus Brockhaus, Sep 10 2002

22 is the smallest number whose base 2 trajectory (A061561) provably does not contain a palindrome. 77 is the next number (cf. A075252) with a completely different trajectory which has this property. 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.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
Interleaving of A176632, 2*A176633, 3*A176634, 12*A176635.
From A.H.M. Smeets, Feb 11 2019: (Start)
Pattern with cycle length 4 in binary representation, represented by contextfree grammars with production rules:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1100010;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 0000101;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101011;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0100000;
the trajectory is similar to that of 22 (see A058042) except for the stopping strings in T_a, T_b, T_c and T_d. (End)

			267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!
Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, 0, 0, -6, 0, 4).

Cf. A061561 (trajectory of 22 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 presumably does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and presumably does not join the trajectory of any term m < n), A092210 (trajectory of n in base 2 presumably does not join the trajectory of any m < n).

Cf. A176632 (a(4*n)), A176633 (a(4*n+1)/2), A176634 (a(4*n+2)/3), A176635 (a(4*n+3)/12).

a075253 n = a075253_list !! n
a075253_list = iterate a055944 77  -- Reinhard Zumkeller, Apr 21 2013

trajectory:=function(init, steps, base) S:=[init]; a:=S[1]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(77, 31, 2);

seq(coeff(series((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8+632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Feb 12 2019

CoefficientList[Series[(77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)), {x,0,40}], x] (* G. C. Greubel, Feb 11 2019 *)
NestWhileList[# + IntegerReverse[#, 2] &, 77,  # !=
IntegerReverse[#, 2] &, 1, 31] (* Robert Price, Oct 18 2019 *)

{m=77; stop=34; c=0; while(c0,d=divrem(k,2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}

((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 11 2019

a(0) = 77; a(1) = 166; a(2) = 267; for n > 2 and
n = 3 (mod 4): a(n) = 48*2^(2*k)-21*2^k where k = (n+5)/4;
n = 0 (mod 4): a(n) = 48*2^(2*k)+33*2^k-3 where k = (n+4)/4;
n = 1 (mod 4): a(n) = 96*2^(2*k)-30*2^k where k = (n+3)/4;
n = 2 (mod 4): a(n) = 96*2^(2*k)+6*2^k-3 where k = (n+2)/4.
G.f.: (77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8 +632*x^9+504*x^10) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
G.f. for the sequence starting at a(3): 3*x^3*(228+299*x-212*x^2 -378*x^3-448*x^4-446*x^5+432*x^6+524*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
a(n+1) = A055944(a(n)). - Reinhard Zumkeller, Apr 21 2013

Three comments added, g.f. edited, MAGMA program and crossrefs added by Klaus Brockhaus, Apr 25 2010

A092211 a(n) = smallest k such that the base-2 Reverse and Add! trajectory of A075252(n) joins the trajectory of k.

Original entry on oeis.org

1, 64, 442, 454, 107, 1066, 1081, 1082, 1085, 1115, 1562, 911, 1070, 266, 3355, 98, 3871, 4099, 4152, 1274, 74, 4202, 4262, 4182, 275, 4633, 4666, 4114, 6166, 6374, 9241, 9466, 8312, 16418, 16490, 16601, 16613, 16616, 298, 16748, 16994, 17002

Offset: 1

Klaus Brockhaus, Feb 25 2004

a(n) <= A075252(n); a(n) = A075252(n) iff the trajectory of A075252(n) does not join the trajectory of any smaller number, i.e., A075252(n) is also a term of A092210.
a(n) determines a 1-1-mapping from the terms of A075252 to the terms of A092210. For the inverse mapping cf. A092212.
Base-2 analog of A089493 (base 10) and A091676 (base 4).

			A075252(1) = 22, the trajectory of 22 (A061561) joins the trajectory of 1 = A092210(1) at 48960, so a(1) = 1. A075252(12) = 1575, the trajectory of 1575 joins the trajectory of 911 = A092210(17) at 184680, so a(12) = 911.

Index entries for sequences related to Reverse and Add!

Cf. A075252, A092210, A092212, A061561, A089493, A091676.

cp's OEIS Frontend

A092212 a(n) = smallest non-palindromic k such that the base-2 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A092210(n).

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A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.

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A092211 a(n) = smallest k such that the base-2 Reverse and Add! trajectory of A075252(n) joins the trajectory of k.

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