A177843 -id:A177843 - cp's OEIS frontend

A077077 Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.

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

Offset: 0

Klaus Brockhaus, Oct 25 2002

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.

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

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).

a077077 n = a077077_list !! n
a077077_list = iterate a055944 775  -- Reinhard Zumkeller, Apr 21 2013

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);

NestWhileList[# + IntegerReverse[#, 2] &, 775,  # !=
IntegerReverse[#, 2] &, 1, 28] (* Robert Price, Oct 18 2019 *)

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);

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

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

A177844 a(n) = 6a(n-1)-8a(n-2) for n > 3; a(0)=279, a(1)=3996, a(2)=16008, a(3)=64784.

Original entry on oeis.org

279, 3996, 16008, 64784, 260640, 1045568, 4188288, 16765184, 67084800, 268387328, 1073645568, 4294774784, 17179484160, 68718706688, 274876366848, 1099508547584, 4398040350720, 17592173723648, 70368719536128

Offset: 0

Klaus Brockhaus, May 14 2010

Related to Reverse and Add trajectory of 775 in base 2: a(n) = A077077(4*n+1)/6, i.e. one sixth of second quadrisection of A077077.

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

Cf. A077077 (Reverse and Add trajectory of 775 in base 2), A177843, A177845, A177846.

[279, 3996] cat [4^(n+5)-47*2^(n+1): n in [2..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(279 + 2322 x - 5736 x^2 + 704 x^3)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)

{m=19; v=concat([279, 3996, 16008, 64784], vector(m-4)); for(n=5, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 4^(n+5)-47*2^(n+1) for n > 1.
G.f.: (279+2322*x-5736*x^2+704*x^3) / ((1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(2): 8*x^2*(2001-3908*x)/((1-2*x)*(1-4*x)).

A177845 a(n) = 6a(n-1)-8a(n-2)-3 for n > 2; a(0)=775, a(1)=8919, a(2)=34223.

Original entry on oeis.org

775, 8919, 34223, 133983, 530111, 2108799, 8411903, 33601023, 134310911, 537057279, 2147856383, 8590680063, 34361229311, 137441935359, 549761777663, 2199035183103, 8796116877311, 35184419799039, 140737583775743

Offset: 0

Klaus Brockhaus, May 14 2010

Related to Reverse and Add trajectory of 775 in base 2: a(n) = A077077(4*n+2)/3, i.e. one third of third quadrisection of A077077.

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

Cf. A077077 (Reverse and Add trajectory of 775 in base 2), A177843, A177844, A177846.

[775] cat [2*4^(n+5)+91*2^(n+2)-1: n in [1..25]]; // Vincenzo Librandi, Sep 24 2013

nxt[{a_,b_}]:={b,6b-8a-3}; Join[{775},Transpose[NestList[nxt,{8919,34223},20]][[1]]] (* or *) Join[{775},LinearRecurrence[{7,-14,8},{8919,34223,133983},20]] (* Harvey P. Dale, Mar 04 2013 *)
CoefficientList[Series[(775 + 3494 x - 17360 x^2 + 13088 x^3)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)

{m=19; v=concat([775, 8919, 34223], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]-3); v}

a(n) = 2*4^(n+5)+91*2^(n+2)-1 for n > 0.
G.f.: (775+3494*x-17360*x^2+13088*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): x*(8919-28210*x+19288*x^2) / ((1-x)*(1-2*x)*(1-4*x)).
a(0)=775, a(1)=8919, a(2)=34223, a(3)=133983, a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3). - Harvey P. Dale, Mar 04 2013

A177846 a(n) = 6a(n-1)-8a(n-2) for n > 2; a(0)=837, a(1)=7896, a(2)=32176.

Original entry on oeis.org

837, 7896, 32176, 129888, 521920, 2092416, 8379136, 33535488, 134179840, 536795136, 2147332096, 8589631488, 34359132160, 137437741056, 549753389056, 2199018405888, 8796083322880, 35184352690176, 140737449558016

Offset: 0

Klaus Brockhaus, May 14 2010

Related to Reverse and Add trajectory of 775 in base 2: a(n) = A077077(4*n+3)/6, i.e. one sixth of fourth quadrisection of A077077.

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

Cf. A077077 (Reverse and Add trajectory of 775 in base 2), A177843, A177844, A177845.

[837] cat [2*4^(n+5)-37*2^(n+2): n in [1..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(837 + 2874 x - 8504 x^2)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)

{m=19; v=concat([837, 7896, 32176], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 2*4^(n+5)-37*2^(n+2) for n > 0.
G.f.: (837+2874*x-8504*x^2) / ((1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): 8*x*(987-1900*x) / ((1-2*x)*(1-4*x)).

cp's OEIS Frontend

A077077 Trajectory of 775 under the Reverse and Add! operation carried out in base 2, written in base 10.

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A177844 a(n) = 6*a(n-1)-8*a(n-2) for n > 3; a(0)=279, a(1)=3996, a(2)=16008, a(3)=64784.

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A177845 a(n) = 6*a(n-1)-8*a(n-2)-3 for n > 2; a(0)=775, a(1)=8919, a(2)=34223.

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Programs

Magma

Mathematica

PARI

Formula

A177846 a(n) = 6*a(n-1)-8*a(n-2) for n > 2; a(0)=837, a(1)=7896, a(2)=32176.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A177844 a(n) = 6a(n-1)-8a(n-2) for n > 3; a(0)=279, a(1)=3996, a(2)=16008, a(3)=64784.

A177845 a(n) = 6a(n-1)-8a(n-2)-3 for n > 2; a(0)=775, a(1)=8919, a(2)=34223.

A177846 a(n) = 6a(n-1)-8a(n-2) for n > 2; a(0)=837, a(1)=7896, a(2)=32176.