A176635 -id:A176635 - cp's OEIS frontend

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

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

A176632 a(n) = 6a(n-1)-8a(n-2)-9 for n > 2; a(0) = 77, a(1) = 897, a(2) = 3333.

Original entry on oeis.org

77, 897, 3333, 12813, 50205, 198717, 790653, 3154173, 12599805, 50365437, 201394173, 805441533, 3221495805, 12885442557, 51540688893, 206160592893, 824638046205, 3298543534077, 13194156834813, 52776592736253

Offset: 0

Klaus Brockhaus, Apr 22 2010

Related to Reverse and Add trajectory of 77 in base 2: a(n) = A075253(4*n), i.e., first quadrisection of A075253.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A075253 (Reverse and Add trajectory of 77 in base 2), A176633, A176634, A176635, A171471.

[77] cat [3*(64*4^n+22*2^n-1): n in [1..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(77 + 358 x - 1868 x^2 + 1424 x^3)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
Join[{77},RecurrenceTable[{a[1]==897,a[2]==3333,a[n]==6a[n-1]-8a[n-2]- 9},a[n],{n,20}]] (* Harvey P. Dale, May 21 2019 *)

{m=20; v=concat([77, 897, 3333], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]-9); v}

a(n) = 3*(64*4^n+22*2^n-1) for n > 0, a(0) = 77.
G.f.: (77+358*x-1868*x^2+1424*x^3)/((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): 3*x*(299-982*x+680*x^2)/((1-x)* (1-2*x)*(1-4*x)).

A176633 a(n) = 6a(n-1)-8a(n-2) for n > 2; a(0) = 83, a(1) = 708, a(2) = 2952.

Original entry on oeis.org

83, 708, 2952, 12048, 48672, 195648, 784512, 3141888, 12575232, 50316288, 201295872, 805244928, 3221102592, 12884656128, 51539116032, 206157447168, 824631754752, 3298530951168, 13194131668992, 52776542404608

Offset: 0

Klaus Brockhaus, Apr 22 2010

nonn

Related to Reverse and Add trajectory of 77 in base 2: a(n) = A075253(4*n+1)/2, i.e., one half of second quadrisection of A075253.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A075253 (Reverse and Add trajectory of 77 in base 2), A176632, A176634, A176635, A171470.

[83] cat [6*(32*4^n-5*2^n): n in [1..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(83 + 210 x - 632 x^2)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{6,-8},{83,708,2952},30] (* Harvey P. Dale, Apr 08 2019 *)

{m=20; v=concat([83, 708, 2952], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 6*(32*4^n-5*2^n) for n > 0, a(1) = 83.
G.f.: (83+210*x-632*x^2)/((1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): 12*x*(59-108*x)/((1-2*x)*(1-4*x)).

A176634 a(n) = 6a(n-1)-8a(n-2)-3 for n > 2; a(0) = 89, a(1) = 519, a(2) = 2063.

Original entry on oeis.org

89, 519, 2063, 8223, 32831, 131199, 524543, 2097663, 8389631, 33556479, 134221823, 536879103, 2147500031, 8589967359, 34359803903, 137439084543, 549756076031, 2199023779839, 8796094070783, 35184374185983

Offset: 0

Klaus Brockhaus, Apr 22 2010

nonn

Related to Reverse and Add trajectory of 77 in base 2: a(n)= A075253(4*n+2)/3, i.e., one third of third quadrisection of A075253.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A075253 (Reverse and Add trajectory of 77 in base 2), A176632, A176633, A176635.

[89] cat [128*4^n+4*2^n-1: n in [1..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(89 - 104 x - 324 x^2 + 336 x^3)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{7,-14,8},{89,519,2063,8223},20] (* Harvey P. Dale, Jun 20 2023 *)

{m=20; v=concat([89, 519, 2063], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]-3); v}

a(n) = 128*4^n+4*2^n-1 for n > 0, a(1) = 89.
G.f.: (89-104*x-324*x^2+336*x^3)/((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): x*(519-1570*x+1048*x^2)/((1-x)* (1-2*x)*(1-4*x)).

A176636 Periodic sequence: Repeat [57, 71].

Original entry on oeis.org

57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57, 71, 57

Offset: 0

Klaus Brockhaus, Apr 22 2010

Continued fraction expansion of (4047+sqrt(16394397))/142.
Inverse binomial transform of 57 followed by 128*A000079.
Third inverse binomial transform of A176635.

Index entries for linear recurrences with constant coefficients, signature (0, 1).

Cf. A000079 (powers of 2), A176635, A176713 (decimal expansion of (4047+sqrt(16394397))/142).

Magma
```
&cat[ [57, 71]: k in [1..35] ];
```

PadRight[{},100,{57,71}] (* Harvey P. Dale, Jun 18 2014 *)

a(n) = 64-7*(-1)^n.
a(n) = -a(n-1)+128 for n > 0; a(0) = 57.
a(n) = a(n-2) for n > 1; a(0) = 57, a(1) = 71.
G.f.: (57+71*x)/((1-x)*(1+x)).

Comment, crossref and keyword cofr added by Klaus Brockhaus, Apr 24 2010

cp's OEIS Frontend

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

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A176632 a(n) = 6*a(n-1)-8*a(n-2)-9 for n > 2; a(0) = 77, a(1) = 897, a(2) = 3333.

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A176633 a(n) = 6*a(n-1)-8*a(n-2) for n > 2; a(0) = 83, a(1) = 708, a(2) = 2952.

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A176634 a(n) = 6*a(n-1)-8*a(n-2)-3 for n > 2; a(0) = 89, a(1) = 519, a(2) = 2063.

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A176636 Periodic sequence: Repeat [57, 71].

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Magma

Mathematica

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Extensions

A176632 a(n) = 6a(n-1)-8a(n-2)-9 for n > 2; a(0) = 77, a(1) = 897, a(2) = 3333.

A176633 a(n) = 6a(n-1)-8a(n-2) for n > 2; a(0) = 83, a(1) = 708, a(2) = 2952.

A176634 a(n) = 6a(n-1)-8a(n-2)-3 for n > 2; a(0) = 89, a(1) = 519, a(2) = 2063.