A075466 -id:A075466 - cp's OEIS frontend

A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

290, 318, 719, 795, 799, 1210, 3903, 4199, 4207, 4219, 4236, 4278, 4279, 4294, 4326, 4333, 4334, 4338, 4402, 4598, 4662, 4726, 5046, 5357, 6157, 6174, 7246, 7247, 7295, 7407, 7549, 8063, 8191, 9211, 12319, 12431, 12463, 12539, 15487, 16519, 16587

Offset: 1

Klaus Brockhaus, Sep 18 2002, revised Jan 28 2004

For 318 (cf. A075153), 266718 (cf. A075466) and 270798 (cf. A075467) one can prove that the base 4 trajectory does not contain a palindrome. A proof for 290 (cf. A075299) has not been found up to now. 4398859679359 is another known candidate (obtained from a remark of David J. Seal, cf. Links) for a term whose trajectory is provably palindrome-free, but is not secured that it does not join the trajectory of some term m < n. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 28 for k < 20000). On the other hand, the trajectories of the terms listed above do not join the trajectory of any smaller term within at least 1000 steps.
Base-4 analog of A063048 (base 10) and A075252 (base 2); subsequence of A075420.
From A.H.M. Smeets, Mar 18 2019: (Start)
David J. Seal (see LINKS) observed a cyclic pattern (length 6) in the trajectories that can be represented by an extended right regular grammar with production rules:
S -> S_a | S_b | S_c | S_d | S_e | S_f,
S_a -> 1033202000232 T_a, T_a -> 222 T_a | 2302333113230
S_b -> 2022321332331 T_b, T_b -> 111 T_b | 1223001203131
S_c -> 10002003002212 T_c, T_c -> 222 T_c | 3221333101333
S_d -> 103312202321111 T_d, T_d -> 111 T_d | 1102023122000
S_e -> 110200123122222 T_e, T_e -> 222 T_e | 2231232001301
S_f -> 213301021321111 T_f, T_f -> 111 T_f | 1113213003312
Within the first 471 terms of this sequence we observed three trajectories with a cyclic pattern (length 6) that can be represented by a context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d | S_e | S_f,
S_a -> 10 T_a 00, T_a -> 3 T_a 0 | T_a0,
S_b -> 11 T_b 01, T_b -> 0 T_b 3 | T_b0,
S_c -> 22 T_c 12, T_c -> 0 T_c 3 | T_c0,
S_d -> 10 T_d 000, T_d -> 3 T_d 0 | T_d0,
S_e -> 11 T_e 301, T_e -> 0 T_e 3 | T_e0,
S_f -> 22 T_f 312, T_f -> 0 T_f 3 | T_f0.
The terminating strings in these context-free grammars are given by:
n     2      359                371
a(n)  318    266718             270798
T_a0  33230  33230000001033230  3323001033230
T_b0  03123  03123010001103123  0312302103123
T_c0  01313  01313120002201313  0131320201313
T_d0  33323  33323000001033323  3332300103323
T_e0  03222  03222301001103222  0322201113222
T_f0  02111  02111312002202111  0211112222111
From the fact that both, right regular grammars and context-free grammars occur, we wonder if other trajectories can be represented by context-sensitive grammars as well, by which other trajectories can be proven never to end up in a palindromic string? (End)

			719 is a term since the trajectory of 719 (presumably) does not lead to an integer which occurs in the trajectory of 290 or of 318.

A.H.M. Smeets, Table of n, a(n) for n = 1..471
Klaus Brockhaus, Illustration: Distribution of terms below 2000000
David J. Seal, Results
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075252, A075420, A075299, A075153, A075466, A075467, A091675.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 4] &, #, # !=IntegerReverse[#, 4] & , 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

318, 1071, 5040, 5985, 10710, 20400, 24225, 43350, 81600, 85425, 165750, 327360, 342705, 664950, 1309440, 1324785, 2629110, 5241600, 5303025, 10524150, 20966400, 21027825, 41973750, 83880960, 84126705, 167925750, 335523840

Offset: 0

Klaus Brockhaus, Sep 05 2002

290 is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome. 318 (not 255 since 255 is a base 4 palindrome) is up to now the smallest number 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.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 3 in {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.

			318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
K. S. Brown, Digit Reversal Sums Leading to Palindromes
Index entries for sequences related to Reverse and Add!

Cf. A058042 (trajectory of binary number 10110 (decimal 22)), A061561 (A058042 written in base 10), A066450 (conjectured minimal k so that the trajectory of k in base n does not lead to a palindrome).
Cf. A075253 (trajectory of 77 in base 2), A075420 (trajectory of n in base 4 (presumably) does not reach a palindrome), A075421 (trajectory of n in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075299 (trajectory of 290 in base 4), A075466 (trajectory of 266718 in base 4), A075467 (trajectory of 270798 in base 4), A076247 (trajectory of 1059774 in base 4), A076248 (trajectory of 1059831 in base 4), A091675 (trajectory of n in base 4 (presumably) does not join the trajectory of any m < n).
Cf. A166912 (a(6*n)/3), A166913 (a(6*n+1)/3), A166914 (a(6*n+2)/240), A166915 (a(6*n+3)/15), A166916 (a(6*n+4)/30), A166917 (a(6*n+5)/240).

trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(318, 26, 4);

NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)

{m=318; stop=29; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0) = 318; a(1) = 1071; for n > 1 and n = 2 (mod 6): a(n) = 5*4^(2*k+5)-5*4^(k+2) where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+5)+55*4^(k+2)-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+5)+30*4^(k+2)-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+5)-5*4^(k+2) where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+5)+235*4^(k+2)-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+5)+150*4^(k+2)-10 where k = (n-7)/6.

G.f.: 3*(106 +357*x +1680*x^2 +1465*x^3 +1785*x^4 -1600*x^5 -1900*x^6 -3400*x^7 -6800*x^8 -9780*x^9 -9860*x^10 +6720*x^11 +10064*x^12 +11088*x^13) / ((1-x)*(1+x+x^2)*(1-2*x^3)*(1+2*x^3)*(1-4*x^3)).

Two comments added, g.f. edited, MAGMA program and cross-references added by Klaus Brockhaus, Oct 26 2009

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

Original entry on oeis.org

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

Offset: 0

Klaus Brockhaus, Sep 18 2002

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.

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

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

Cf. A075153, A075421, A075466.

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

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

a(0), ..., a(10) as above; for n > 10 and n = 5 (mod 6): a(n) = 5*4^(2*k+10)+15341035*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)+14361820*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))

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

Original entry on oeis.org

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

Offset: 0

Klaus Brockhaus, Oct 03 2002

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.

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

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

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

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

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

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))

A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

1059831, 4728312, 7831065, 14433270, 24913965, 56412450, 92165625, 208908750, 396926625, 710289750, 1336954560, 1398889905, 2715199350, 5363547840, 5614238385, 10894222710, 21453945600, 21701687025, 43073052150

Offset: 0

Klaus Brockhaus, Oct 03 2002

1059831 = A075421(1105 ) is the fifth 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.

			1059831 (decimal) = 10002233313 -> 10002233313 + 31333220001 = 102002113320 = 4728312 (decimal).

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

Cf. A075421, A075153, A075466, A075467, A076247.

NestWhileList[# + IntegerReverse[#, 4] &, 1059831,  # != IntegerReverse[ #, 4] &, 1, 23] (* Robert Price, Oct 19 2019 *)

{m=1059831; stop=19; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0), ..., a(7) as above; for n > 7 and n = 2 (mod 6): a(n) = 5*4^(2*k+9)+3836395*4^k-15 where k = (n+4)/6; n = 3 (mod 6): a(n) = 10*4^(2*k+9)+2450070*4^k-10 where k = (n+3)/6; n = 4 (mod 6): a(n) = 20*4^(2*k+9)-326420*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+9)+3544540*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 40*4^(2*k+9)+1927800*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 80*4^(2*k+9)-322580*4^k where k = (n-1)/6. G.f.: -3*(668508000*x^19+444361200*x^18+222142800*x^17-528080680*x^16-356464620*x^15 -125753060*x^14-299532884*x^13-188180432*x^12-143040640*x^11+128992350*x^10+90219415*x^9 +38288125*x^8+28112975*x^7+6666425*x^6+5752375*x^5+424135*x^4+3044705*x^3+2610355*x^2 + 1576104*x+353277)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A077408 Trajectory of 103 under the Reverse and Add! operation carried out in base 3, written in base 10.

Original entry on oeis.org

103, 230, 436, 776, 2424, 3856, 7400, 20856, 30928, 60920, 220248, 242704, 432896, 857152, 1460408, 2754688, 5134016, 16206744, 24437488, 44623424, 138104472, 201737128, 401511824, 1438324704, 1601682040, 2820726320, 5622321088

Offset: 0

Klaus Brockhaus, Nov 05 2002

103 = A077405(0) is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 3 does not lead to a palindrome. Its trajectory does not exhibit any recognizable regularity, so that the method by which the base-2 trajectories of 22 (cf. A061561), 77 (cf. A075253), 442 (cf. A075268) etc. as well as the base-4 trajectories of 318 (cf. A075153), 266718 (cf. A075466), 270798 (cf. A075467) etc. can be proved to be palindrome-free (cf. Links), is not applicable here.

			103 (decimal) = 10211 -> 10211 + 11201 = 22112 = 230 (decimal).

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

Cf. A058042, A077405, A066450, A061561, A075253, A075268, A075153, A075466, A075467.

m := 103; stop := 28; c := 0; while c < stop do write(m:group(0),","); k := m; rev := 0; while k > 0 do rev := 3*rev + (k mod 3); k := k div 3; end; inc(c); m := m+rev; end;

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A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

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

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

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A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A077408 Trajectory of 103 under the Reverse and Add! operation carried out in base 3, written in base 10.

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