A177684 -id:A177684 - cp's OEIS frontend

A077076 Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.

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

Offset: 0

Klaus Brockhaus, Oct 25 2002

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.

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

a077076 n = a077076_list !! n
a077076_list = iterate a055944 537  -- 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(537, 27, 2);

NestWhileList[# + IntegerReverse[#, 2] &, 537,  # !=
IntegerReverse[#, 2] &, 1, 27] (* 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(537,27);

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

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

A177682 a(n) = 6a(n-1)-8a(n-2)-3 for n > 4; a(0)=179, a(1)=1571, a(2)=12511, a(3)=155403, a(4)=572951.

Original entry on oeis.org

179, 1571, 12511, 155403, 572951, 2194479, 8583263, 33943743, 134996351, 538428159, 2150598143, 8596163583, 34372196351, 137463869439, 549805645823, 2199122919423, 8796292349951, 35184770744319, 140738285666303

Offset: 0

Klaus Brockhaus, May 12 2010

Related to Reverse and Add trajectory of 537 in base 2: a(n) = A077076(4*n)/3, i.e. one third of first quadrisection of A077076.

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

Cf. A077076 (Reverse and Add trajectory of 537 in base 2), A177683, A177684, A177685.

[179, 1571, 12511] cat [2*4^(n+5)+6083*2^(n-1)-1: n in [3..25]]; // Vincenzo Librandi, Sep 24 2013

Join[{179,1571,12511},LinearRecurrence[{7,-14,8},{155403,572951,2194479},20]] (* Harvey P. Dale, Oct 06 2011 *)
CoefficientList[Series[(179 + 318 x + 4020 x^2 + 88388 x^3 - 352284 x^4 + 259376 x^5)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)

{m=19; v=concat([179, 1571, 12511, 155403, 572951], vector(m-5)); for(n=6 ,m, v[n]=6*v[n-1]-8*v[n-2]-3); v}

a(n) = 2*4^(n+5)+6083*2^(n-1)-1 for n > 2.
G.f.: (179+318*x+4020*x^2+88388*x^3-352284*x^4+259376*x^5) / ((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(3): x^3*(155403-514870*x+359464*x^2) / ((1-x)*(1-2*x)*(1-4*x)).

A177683 a(n) = 6a(n-1)-8a(n-2) for n > 4; a(0)=191, a(1)=1587, a(2)=14161, a(3)=123004, a(4)=508152.

Original entry on oeis.org

191, 1587, 14161, 123004, 508152, 2064880, 8324064, 33425344, 133959552, 536354560, 2146450944, 8587869184, 34355607552, 137430691840, 549739290624, 2198990209024, 8796026929152, 35184239902720, 140737223983104

Offset: 0

Klaus Brockhaus, May 12 2010

Related to Reverse and Add trajectory of 537 in base 2: a(n) = A077076(4*n+1)/6, i.e., one sixth of second quadrisection of A077076.

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

Cf. A077076 (Reverse and Add trajectory of 537 in base 2), A177682, A177684, A177685.

[191, 1587, 14161] cat [2*4^(n+5)-2017*2^(n-1): n in [3..25]]; // Vincenzo Librandi, Sep 24 2013

Join[{191,1587,14161},Transpose[NestList[{Last[#],6Last[#]-8First[#]}&,{123004,508152},20]][[1]]]  (* Harvey P. Dale, Mar 06 2011 *)
CoefficientList[Series[(191 + 441 x + 6167 x^2 + 50734 x^3 - 116584 x^4)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{6,-8},{191,1587,14161,123004,508152},20] (* Harvey P. Dale, Oct 16 2019 *)

{m=19; v=concat([191, 1587, 14161, 123004, 508152], vector(m-5)); for(n=6 ,m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 2*4^(n+5)-2017*2^(n-1) for n > 2.
G.f.: (191+441*x+6167*x^2+50734*x^3-116584*x^4) / ((1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(3): 4*x^3*(30751-57468*x) / ((1-2*x)*(1-4*x)).

A177685 a(n) = 6a(n-1)-8a(n-2) for n > 4; a(0)=603, a(1)=4731, a(2)=58834, a(3)=254204, a(4)=1032696.

Original entry on oeis.org

603, 4731, 58834, 254204, 1032696, 4162544, 16713696, 66981824, 268181376, 1073233664, 4293950976, 17177836544, 68715411456, 274869776384, 1099495366656, 4398013988864, 17592120999936, 70368614088704

Offset: 0

Klaus Brockhaus, May 12 2010

nonn

Related to Reverse and Add trajectory of 537 in base 2: a(n) = A077076(4*n+3)/6, i.e., one sixth of fourth quadrisection of A077076.

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

Cf. A077076 (Reverse and Add trajectory of 537 in base 2), A177682, A177683, A177684.

[603, 4731, 58834] cat [4^(n+6)-1985*2^(n-1): n in [3..25]]; // Vincenzo Librandi, Sep 24 2013

CoefficientList[Series[(603 + 1113 x + 35272 x^2 - 60952 x^3 - 21856 x^4)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{6,-8},{603,4731,58834,254204,1032696},20] (* Harvey P. Dale, May 26 2019 *)

{m=18; v=concat([603, 4731, 58834, 254204, 1032696], vector(m-5)); for(n=6 ,m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 4^(n+6)-1985*2^(n-1) for n > 2.
G.f.: (603+1113*x+35272*x^2-60952*x^3-21856*x^4) / ((1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(3): 4*x^3*(63551-123132*x) / ((1-2*x)*(1-4*x)).

cp's OEIS Frontend

A077076 Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.

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A177682 a(n) = 6*a(n-1)-8*a(n-2)-3 for n > 4; a(0)=179, a(1)=1571, a(2)=12511, a(3)=155403, a(4)=572951.

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Magma

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A177683 a(n) = 6*a(n-1)-8*a(n-2) for n > 4; a(0)=191, a(1)=1587, a(2)=14161, a(3)=123004, a(4)=508152.

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Magma

Mathematica

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A177685 a(n) = 6*a(n-1)-8*a(n-2) for n > 4; a(0)=603, a(1)=4731, a(2)=58834, a(3)=254204, a(4)=1032696.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A177682 a(n) = 6a(n-1)-8a(n-2)-3 for n > 4; a(0)=179, a(1)=1571, a(2)=12511, a(3)=155403, a(4)=572951.

A177683 a(n) = 6a(n-1)-8a(n-2) for n > 4; a(0)=191, a(1)=1587, a(2)=14161, a(3)=123004, a(4)=508152.

A177685 a(n) = 6a(n-1)-8a(n-2) for n > 4; a(0)=603, a(1)=4731, a(2)=58834, a(3)=254204, a(4)=1032696.