A061561 -id:A061561

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A058042

Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.

+10
21

10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

According to J. Walker, Ronald Sprague has proved that this trajectory does not contain a palindrome. [I would like a reference for this.] Another proof has been given by Klaus Brockhaus.

10110 is the smallest number with this property in base 2. The analogous number in base 10 is believed to be 196, but its trajectory (see A006960) has never been proved not to contain a palindrome.

The binary numbers have a regular pattern with cycle length 4:

a(4k) = 101^(k+1)010^(k+1) for k >= 1,

a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,

a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,

a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation. - A.H.M. Smeets, Feb 03 2019

From A.H.M. Smeets, Feb 11 2019: (Start)

Pattern with cycle length 4 represented by contextfree grammars with production rules:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;

see also A058042 for similar grammars for the binary represented trajectory of 77. (End)

LINKS

T. D. Noe, Table of n, a(n) for n = 0..500

T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing, Aug 22 1995.

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2

David J. Seal, Proofs similar to base 2 for base 4, 11, 17 and 26

J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest

Index entries for sequences related to Reverse and Add!

FORMULA

a(n) = A007088(A061561(n)). - Reinhard Zumkeller, Apr 21 2013

MATHEMATICA

Clear[a]; a[0] = 10110; a[n_] := a[n] = (m = IntegerDigits[ a[n-1] ]; m2 = FromDigits[m, 2]; IntegerDigits[ FromDigits[m // Reverse, 2] + m2, 2] // FromDigits); Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 03 2013 *)

PROG

(ARIBAS) var m, c, rev: integer; end; m := 22; c := 1; bit_write(m); write(" "); rev := bit_reverse(m); while m <> rev and c < 25 do inc(c); m := m + rev; bit_write(m); write(" "); rev := bit_reverse(m); end;

(Haskell)

a058042 = a007088 . a061561 -- Reinhard Zumkeller, Apr 21 2013

CROSSREFS

See A061561 for the terms of A058042 written in base 10. Cf. A016016, A006960, A023108.

KEYWORD

nonn,nice,base

AUTHOR

N. J. A. Sloane, May 18 2001

EXTENSIONS

More terms from Klaus Brockhaus, May 27 2001

STATUS

approved

A075252

Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

+10
17

22, 77, 442, 537, 775, 1066, 1081, 1082, 1085, 1115, 1562, 1575, 1587, 2173, 3355, 3599, 3871, 4099, 4153, 4185, 4193, 4202, 4262, 4285, 4402, 4633, 4666, 6163, 6166, 6374, 9241, 9466, 16544, 16546, 16586, 16601, 16613, 16616, 16720, 16748, 16994

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Base-2 analog of A063048 (base 10) and A075421 (base 4); subsequence of A066059. - For the trajectory of 22 (cf. A061561) and the trajectory of 77 (cf. A075253) it has been proved that they do not contain a palindrome. A similar proof can be given for most terms of this sequence, but there are a few terms (4262, 17498, 33378, 33898, ...) whose trajectory does not show the kind of regularity that can be utilized for the construction of a proof. - 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 84 for numbers < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.

From A.H.M. Smeets, Feb 12 2019: (Start)

Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).

A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);

A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)

Decimal representations for 10 A_a(n) 00 are given by A306514(n).

Binary representations for 10 A_a(n) 00 are given by A306515(n).

(End)

LINKS

Table of n, a(n) for n=1..41.

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2

Index entries for sequences related to Reverse and Add!

EXAMPLE

442 is a term since the trajectory of 442 (presumably) does not lead to an integer which occurs in the trajectory of 22 or of 77.

MATHEMATICA

limit = 10^2; (* Assumes that there is no palindrome if none is found before "limit" iterations *)

utraj = {};

Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # != IntegerReverse[#, 2] & , 1, limit];

If[Length[x] >= limit && Intersection[x, utraj] == {},

utraj = Union[utraj, x]; True,

utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

CROSSREFS

Cf. A063048, A075421, A066059, A058042, A061561, A075253, A306514, A306515.

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Sep 10 2002

STATUS

approved

A066059

Integers such that the 'Reverse and Add!' algorithm in base 2 (cf. A062128) does not lead to a palindrome.

+10
15

22, 26, 28, 35, 37, 41, 46, 47, 49, 60, 61, 67, 75, 77, 78, 84, 86, 89, 90, 94, 95, 97, 105, 106, 108, 110, 116, 120, 122, 124, 125, 131, 135, 139, 141, 147, 149, 152, 155, 157, 158, 163, 164, 166, 169, 172, 174, 177, 180, 182, 185, 186, 190, 191, 193, 197, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The analog of A023108 in base 2.

It seems that for all these numbers it can be proven that they never reach a palindrome. For this it is sufficient to prove this for all seeds as given in A075252. As observed, for all numbers in A075252, lim_{n -> inf} t(n+1)/t(n) is 1 or 2 (1 for even n, 2 for odd n or reverse); i.e., lim_{n -> inf} t(n+2)/t(n) = 2, t(n) being the n-th term of the trajectory. - A.H.M. Smeets, Feb 10 2019

LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Klaus Brockhaus, On the'Reverse and Add!' algorithm in base 2

Index entries for sequences related to Reverse and Add!

MATHEMATICA

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)

Select[Range[200],

Length@NestWhileList[# + IntegerReverse[#, 2] &, #, # !=

IntegerReverse[#, 2] &, 1, limit] == limit + 1 &] (* Robert Price, Oct 14 2019 *)

PROG

(ARIBAS): For function b2reverse see A066057; function a066059(mx, stop: integer); var k, c, m, rev: integer; begin for k := 1 to mx do c := 0; m := k; rev := b2reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := b2reverse(m); end; if c >= stop then write(k, " "); end; end; end; a066059(210, 300).

CROSSREFS

Cf. A062128, A023108, A062130, A033865, A058042, A061561, A066057.

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Dec 04 2001

STATUS

approved

A075153

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..26.

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!

FORMULA

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)).

EXAMPLE

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

MATHEMATICA

NestWhileList[# + IntegerReverse[#, 4] &, 318, # !=

IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)

PROG

(PARI) {m=318; stop=29; c=0; while(c<stop, print1(k=m, ", "); rev=0; while(k>0, d=divrem(k, 4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

(Magma) 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);

CROSSREFS

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).

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Sep 05 2002

EXTENSIONS

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

STATUS

approved

A075253

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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)

LINKS

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).

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) {m=77; stop=34; c=0; while(c<stop, print1(k=m, ", "); rev=0; while(k>0, d=divrem(k, 2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}

(Magma) 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);

(Haskell)

a075253 n = a075253_list !! n

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

(Sage) ((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

CROSSREFS

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).

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Sep 10 2002

EXTENSIONS

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

STATUS

approved

A171472

a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 7, a(1) = 30.

+10
13

7, 30, 124, 504, 2032, 8160, 32704, 130944, 524032, 2096640, 8387584, 33552384, 134213632, 536862720, 2147467264, 8589901824, 34359672832, 137438822400, 549755551744, 2199022731264, 8796091973632, 35184369991680

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Related to Reverse and Add trajectory of 22 in base 2: A061561(4*n+2) = 12*a(n).

Third binomial transform of A010729.

a(n) in base 2 is n+3 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (6,-8).

FORMULA

a(n) = 8*4^n-2^n.

G.f.: (7-12*x)/((1-2*x)*(1-4*x)).

a(n) = A171499(n+1)/2. - Hussam al-Homsi, Jun 06 2021

E.g.f.: exp(2*x)*(8*exp(2*x) - 1). - Stefano Spezia, Sep 27 2023

MATHEMATICA

LinearRecurrence[{6, -8}, {7, 30}, 30] (* Harvey P. Dale, Sep 01 2016 *)

PROG

(PARI) {m=22; v=concat([7, 30], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

(Magma) [8*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, May 31 2011

CROSSREFS

Cf. A061561, A010729 (repeat 7, 9), A171470, A171471, A171473, A171499.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Dec 09 2009

STATUS

approved

A171473

a(n) = 6*a(n-1) - 8*a(n-2)-3 for n > 1; a(0) = 35, a(1) = 135.

+10
12

35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Related to Reverse and Add trajectory of 22 in base 2: A061561(4*n+3) = 3*a(n).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

FORMULA

a(n) = 32*4^n + 4*2^n - 1.

G.f.: (35-110*x+72*x^2)/((1-x)*(1-2*x)*(1-4*x)).

a(n) = A092431(n+3).

a(n+1) - a(n) = A049775(n+5).

E.g.f.: exp(x)*(32*exp(3*x) + 4*exp(x) - 1). - Stefano Spezia, Sep 27 2023

PROG

(PARI) {m=20; v=concat([35, 135], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]-3); v}

(Magma) [32*4^n+4*2^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

CROSSREFS

Cf. A061561, A092431, A049775, A171470, A171471, A171472.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Dec 09 2009

STATUS

approved

A075268

Trajectory of 442 under the Reverse and Add! operation carried out in base 2.

+10
11

442, 629, 1326, 2259, 5508, 6585, 11628, 15129, 24912, 26259, 52038, 77337, 155394, 221931, 442374, 639009, 1179738, 1917027, 3539130, 5062869, 10666542, 18285939, 45369156, 54513657, 96444396, 125792217, 207562704, 220034931

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

22, 77 and 442 are the first terms of A075252. The base 2 trajectory of 442 is completely different from the trajectories of 22 (cf. A061561) and 77 (cf. A075253). Using the formula given below one can prove that it does not contain a palindrome.

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 2*A177420, A177421, 6*A177422, 3*A177423.

LINKS

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!

FORMULA

a(0), ..., a(28) as above; a(29) = 703932681; a(30) =1310348526; a(31) = 2309980455; a(32) = 6143702712; a(33) = 7131271077; a(34) = 12699398352; a(35) = 13441412493; for n > 35 and

n = 0 (mod 4): a(n) = 3*2^(2*k+23)-12576771*2^k where k = (n-16)/4;

n = 1 (mod 4): a(n) = 3*2^(2*k+23)+12576771*2^k-3 where k = (n-17)/4;

n = 2 (mod 4): a(n) = 6*2^(2*k+23)-12576771*2^k where k = (n-18)/4;

n = 3 (mod 4): a(n) = 6*2^(2*k+23)+37730313*2^k-3 where k = (n-19)/4.

G.f.: (442+629*x+372*x^3+1530*x^4-192*x^5-2244*x^6-852*x^7-3784*x^8-8090*x^9 +5046*x^10+29034*x^11+47016*x^12+54354*x^13+79152*x^14+70254*x^15+65196*x^16 +358986*x^17+724128*x^18+334026*x^19+2081820*x^20+6043662*x^21+18678462*x^22+8601966*x^23 -23147244*x^24-15039648*x^25 -31927752*x^26-67877562*x^27+43880046*x^28+297766074*x^29 +396480108*x^30+734881086*x^31+3072255774*x^32+1018370430*x^33-3939844260*x^34-4608944376*x^35 -6616834356*x^36-3107825028*x^37+6655931736*x^38+7777900872*x^39+484428384*x^40 -2233413600*x^41-62899200*x^42+188697600*x^43) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).

G.f. for the sequence starting at a(36): 3*x^36*(8455782368+8724086815*x -8321630144*x^2-8589934590*x^3-17045716960*x^4-18118934750*x^5+16911564736*x^6 +17984782524*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

EXAMPLE

442 (decimal) = 110111010 -> 110111010 + 010111011 = 1001110101 = 629 (decimal).

MATHEMATICA

NestWhileList[# + IntegerReverse[#, 2] &, 442, # !=

IntegerReverse[#, 2] &, 1, 27] (* Robert Price, Oct 18 2019 *)

PROG

(PARI) trajectory(n, steps) = {local(v, k=n); for(j=0, steps, print1(k, ", "); v=binary(k); k+=sum(j=1, #v, 2^(j-1)*v[j]))};

trajectory(442, 28);

(Haskell)

a075268 n = a075268_list !! n

a075268_list = iterate a055944 442 -- Reinhard Zumkeller, Apr 21 2013

CROSSREFS

Cf. A058042 (trajectory of 22 in base 2, written in base 2), A061561 (trajectory of 22 in base 2), A075253 (trajectory of 77 in base 2), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n).

Cf. A177420 (a(4*n)/2), A177421 (a(4*n+1)), A177422 (a(4*n+2)/6), A177423 (a(4*n+3)/3).

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Sep 11 2002

EXTENSIONS

Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 07 2010

STATUS

approved

A077076

Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.

+10
9

537, 1146, 1899, 3618, 4713, 9522, 14427, 28386, 37533, 84966, 138123, 353004, 466209, 738024, 833301, 1525224, 1718853, 3048912, 3239469, 6196176, 6583437, 12389280, 12770397, 24975264, 25749789, 49944384, 50706621, 100282176

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The base 2 trajectory of 537 = A075252(4) provably does not contain a palindrome. A proof can be based on the formula given below.

lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.

lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.

Interleaving of 3*A177682, 6*A177683, 3*A177684, 6*A177685.

LINKS

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!

FORMULA

a(0), ..., a(11) as above; for n > 11 and

n = 0 (mod 4): a(n) = 3*2^(2*k+13)+18249*2^k-3 where k = (n-4)/4;

n = 1 (mod 4): a(n) = 6*2^(2*k+13)-12102*2^k where k = (n-5)/4;

n = 2 (mod 4): a(n) = 6*2^(2*k+13)+11718*2^k-3 where k = (n-6)/4;

n = 3 (mod 4): a(n) = 12*2^(2*k+13)-11910*2^k where k = (n-7)/4.

G.f.: 3*(179+382*x+96*x^2+60*x^3-328*x^4-444*x^5+1170*x^6+2232*x^7 +1166*x^8+5644*x^9+15402*x^10+46922*x^11+39850*x^12-62920*x^13-132612*x^14 -97532*x^15-34148*x^16+83800*x^17+109224*x^18+21856*x^19) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).

G.f. for the sequence starting at a(12): 3*x^12*(155403+246008*x-188442*x^2-229616*x^3-260350*x^4-508920*x^5+293388*x^6+492528*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

EXAMPLE

537 (decimal) = 1000011001 -> 1000011001 + 1001100001 = 10001111010= 1146 (decimal).

MATHEMATICA

NestWhileList[# + IntegerReverse[#, 2] &, 537, # !=

IntegerReverse[#, 2] &, 1, 27] (* Robert Price, Oct 18 2019 *)

PROG

(PARI) trajectory(n, steps) = {local(v, k=n); for(j=0, steps, print1(k, ", "); v=binary(k); k+=sum(j=1, #v, 2^(j-1)*v[j]))};

trajectory(537, 27);

(Haskell)

a077076 n = a077076_list !! n

a077076_list = iterate a055944 537 -- Reinhard Zumkeller, Apr 21 2013

CROSSREFS

Cf. A058042 (trajectory of 22 in base 2, written in base 2), A061561 (trajectory of 22 in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077077 (trajectory of 775 in base 2), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n).

Cf. A177682 (a(4*n)/3), A177683 (a(4*n+1)/6), A177684 (a(4*n+2)/3), A177685 (a(4*n+3)/6).

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Oct 25 2002

EXTENSIONS

Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 12 2010

STATUS

approved

A077077

Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.

+10
9

775, 1674, 2325, 5022, 8919, 23976, 26757, 47376, 49581, 96048, 102669, 193056, 197469, 388704, 401949, 779328, 788157, 1563840, 1590333, 3131520, 3149181, 6273408, 6326397, 12554496, 12589821, 25129728, 25235709, 50274816, 50345469

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The base 2 trajectory of 775 = A075252(5) provably does not contain a palindrome. A proof can be based on the formula given below.

lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.

lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.

Interleaving of A177843, 6*A177844, 3*A177845, 6*A177846.

LINKS

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!

FORMULA

a(0), ..., a(5) as above; for n > 5 and

n = 2 (mod 4): a(n) = 3*2^(2*k+7)+273*2^k-3 where k = (n+6)/4;

n = 3 (mod 4): a(n) = 6*2^(2*k+7)-222*2^k where k = (n+5)/4;

n = 0 (mod 4): a(n) = 6*2^(2*k+7)+54*2^k-3 where k = (n+4)/4;

n = 1 (mod 4): a(n) = 12*2^(2*k+7)-282*2^k where k = (n+3)/4.

a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)+2*a(n-4)+2*a(n-5)-4*a(n-6)-4*a(n-7)-3 for n > 12; a(0), ..., a(12) as above.

G.f.: (775+1674*x+1944*x^4+8910*x^5+4650*x^6-14508*x^7-19840*x^8-22644*x^9- 1860*x^10+28680*x^11+14328*x^12-2112*x^13) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).

G.f. for the sequence starting at a(6): 3*(8919+15792*x-10230*x^2- 15360*x^3-15358*x^4-31696*x^5+16668*x^6+31264*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

EXAMPLE

775 (decimal) = 1100000111 -> 1100000111 + 1110000011 = 11010001010 = 1674 (decimal).

MATHEMATICA

NestWhileList[# + IntegerReverse[#, 2] &, 775, # !=

IntegerReverse[#, 2] &, 1, 28] (* Robert Price, Oct 18 2019 *)

PROG

(PARI) trajectory(n, steps) = {local(v, k=n); for(j=0, steps, print1(k, ", "); v=binary(k); k+=sum(j=1, #v, 2^(j-1)*v[j]))};

trajectory(775, 28);

(Magma) 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(775, 28, 2);

(Haskell)

a077077 n = a077077_list !! n

a077077_list = iterate a055944 775 -- Reinhard Zumkeller, Apr 21 2013

CROSSREFS

Cf. A058042 (trajectory of 22 in base 2, written in base 2), A061561 (trajectory of 22 in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n).

Cf. A177843 (a(4*n)), A177844 (a(4*n+1)/6), A177845 (a(4*n+2)/3), A177846 (a(4*n+3)/6).

KEYWORD

base,nonn

AUTHOR

Klaus Brockhaus, Oct 25 2002

EXTENSIONS

Comment edited, three comments and formula added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by Klaus Brockhaus, May 14 2010

STATUS

approved

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