cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Klaus Brockhaus, Sep 05 2002

Keywords

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.

Examples

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

Links

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).

Programs

  • 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);
  • Mathematica
    NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)
  • PARI
    {m=318; stop=29; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

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)).

Extensions

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

A166912 a(n) = 20*a(n-1) - 64*a(n-2) - 225 for n > 2; a(0) = 106, a(1) = 8075, a(2) = 114235.

Original entry on oeis.org

106, 8075, 114235, 1767675, 28042235, 447713275, 7159562235, 114537594875, 1832539914235, 29320392212475, 469125289738235, 7506000693166075, 120095995320074235, 1921535862038855675, 30744573540292362235
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n) = 3*a(n).
lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A075153, A166913, A166914, A166915, A166916, A166917, A167120, A167121, A167122.

Programs

  • Mathematica
    Join[{106},LinearRecurrence[{21,-84,64},{8075,114235,1767675},20]] (* Harvey P. Dale, Jun 07 2012 *)
  • PARI
    m=15; v=concat([106, 8075, 114235], vector(m-3)); for(n=4, m, v[n]=20*v[n-1]-64*v[n-2]-225); v

Formula

a(n) = (1280*16^n + 940*4^n - 15)/3 for n > 0.
G.f.: (106 + 5849*x - 46436*x^2 + 40256*x^3)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 28 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/3)*(-15*exp(x) + 940*exp(4*x) + 1280*exp(16*x)) - 629. (End)

A166913 a(n) = 20*a(n-1) - 64*a(n-2) - 150 for n > 2; a(0) = 357, a(1) = 14450, a(2) = 221650.

Original entry on oeis.org

357, 14450, 221650, 3508050, 55975250, 894989650, 14317376850, 229068199250, 3665051866450, 58640672576850, 938250132084050, 15011999596762450, 240191983481869650, 3843071695444596050, 61489146966052263250
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+1) = 3*a(n).
lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A075153, A166912, A166914, A166915, A166916, A166917, A167120, A167121, A167122.

Programs

  • Mathematica
    Join[{357},LinearRecurrence[{21,-84,64},{14450,221650,3508050},20]] (* Harvey P. Dale, Jun 18 2014 *)
  • PARI
    m=15; v=concat([357, 14450, 221650], vector(m-3)); for(n=4, m, v[n]=20*v[n-1]-64*v[n-2]-150); v

Formula

a(n) = (2560*16^n + 600*4^n - 10)/3 for n > 0.
G.f.: (357 + 6953*x - 51812*x^2 + 44352*x^3)/((1-x)*(1-4*x)*(1-16*x)).
a(0)=357, a(1)=14450, a(2)=221650, a(3)=3508050, a(n)=21*a(n-1)- 84*a(n-2)+ 64*a(n-3). - Harvey P. Dale, Jun 18 2014
E.g.f.: (1/3)*(-10*exp(x) + 600*exp(4*x) + 2560*exp(16*x)) - 693. - G. C. Greubel, May 28 2016

A166916 a(n) = 20*a(n-1) - 64*a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

Original entry on oeis.org

357, 5525, 87637, 1399125, 22373717, 357930325, 5726688597, 91626231125, 1466016552277, 23456252253525, 375299985724757, 6004799570269525, 96076792319006037, 1537228673882871125, 24595658769241036117
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+4) = 30*a(n).
lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A075153, A166912, A166913, A166914, A166915, A166917, A006105, A167031, A167032, A167033.

Programs

  • Mathematica
    LinearRecurrence[{21,-84,64},{357,5525,87637},20] (* Harvey P. Dale, Sep 24 2012 *)
  • PARI
    m=15; v=concat([357, 5525], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]-15); v

Formula

a(n) = (1024*16^n + 48*4^n - 1)/3.
G.f.: (357 - 1972*x + 1600*x^2)/((1-x)*(1-4*x)*(1-16*x)).
a(0)=357, a(1)=5525, a(2)=87637, a(n)=21*a(n-1)-84*a(n-2)+64*a(n-3). - Harvey P. Dale, Sep 24 2012
E.g.f.: (1/3)*(1024*exp(16*x) + 48*exp(4*x) - exp(x)). - G. C. Greubel, May 28 2016

A166914 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 21, a(1) = 340.

Original entry on oeis.org

21, 340, 5456, 87360, 1398016, 22369280, 357912576, 5726617600, 91625947136, 1466015416320, 23456247709696, 375299967549440, 6004799497568256, 96076792028200960, 1537228672719650816, 24595658764588154880
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+2) = 240*a(n).

Links

Crossrefs

Cf. A075153, A166912, A166913, A166915, A166916, A166917, A166927, A166965, A166984.

Programs

  • Magma
    [Binomial(4^(n+3), 2)/96: n in [0..30]]; // G. C. Greubel, Oct 02 2024
  • Mathematica
    CoefficientList[Series[(21-80x)/((1-4x)(1-16x)),{x,0,20}],x]  (* or *) LinearRecurrence[{20,-64},{21,340},20] (* Harvey P. Dale, Feb 23 2011 & Mar 30 2012 *)
  • PARI
    {m=15; v=concat([21, 340], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}
  • SageMath
    A166914=BinaryRecurrenceSequence(20,-64,21,340)
[A166914(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = (64*16^n - 4^n)/3.
G.f.: (21 - 80*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
From G. C. Greubel, May 28 2016: (Start)
a(n) = 20*a(n-1) - 64*a(n-2).
E.g.f.: (1/3)*(-exp(4*x) + 64*exp(16*x)). (End)

A166917 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 85, a(1) = 1364.

Original entry on oeis.org

85, 1364, 21840, 349504, 5592320, 89478144, 1431654400, 22906486784, 366503854080, 5864061927424, 93824991887360, 1501199874392064, 24019198007050240, 384307168179912704, 6148914691147038720, 98382635059426361344, 1574122160955116748800, 25185954575299047849984
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+5) = 240*a(n).

Links

Crossrefs

Cf. A075153, A166912, A166913, A166914, A166915, A166916, A166927, A166965, A166984.

Programs

  • Magma
    [Binomial(4^(n+4), 2)/384: n in [0..30]]; // G. C. Greubel, Oct 02 2024
  • Mathematica
    LinearRecurrence[{20,-64}, {85, 1364}, 50] (* G. C. Greubel, May 28 2016 *)
  • PARI
    {m=15; v=concat([85, 1364], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}
  • SageMath
    A166917=BinaryRecurrenceSequence(20,-64,85,1364)
[A166917(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = (256*16^n - 4^n)/3.
G.f.: (85 - 336*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
E.g.f.: (1/3)*(256*exp(16*x) - exp(4*x)). - G. C. Greubel, May 28 2016

A167031 a(n) = 20*a(n-1) - 64*a(n-2) + 1 for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 337, 5461, 87653, 1403557, 22461349, 359399333, 5750460325, 92007649189, 1472123522981, 23553980911525, 376863712759717, 6029819476856741, 96477111920512933, 1543633791891427237, 24698140674915717029
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A166915, A166916, A006105, A167032, A167033.

Programs

  • Magma
    [ n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2)+1: n in [1..17] ];
  • Mathematica
    LinearRecurrence[{21, -84, 64}, {1, 20, 337}, 50] (* G. C. Greubel, May 30 2016 *)

Formula

a(n) = (241*16^n - 65*4^n + 4)/180.
G.f.: (1-x+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-2).
E.g.f.: (1/180)*(241*exp(16*x) - 65*exp(4*x) + 4*exp(x)). (End)

A167032 a(n) = 20*a(n-1) - 64*a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 21.

Original entry on oeis.org

1, 21, 358, 5818, 93450, 1496650, 23952202, 383258442, 6132227914, 98116017994, 1569857773386, 25117730316106, 401883708825418, 6430139436277578, 102882231360724810, 1646115703292731210, 26337851258768236362
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A006105, A166915, A166916, A167031, A167033.

Programs

  • Magma
    [ n le 2 select 20*n-19 else 20*Self(n-1)-64*Self(n-2)+2: n in [1..17] ];
  • Maple
    A167032:=n->(257*16^n - 85*4^n + 8)/180: seq(A167032(n), n=0..25); # Wesley Ivan Hurt, May 30 2016
  • Mathematica
    LinearRecurrence[{21, -84, 64}, {1,21,358}, 50] (* G. C. Greubel, May 30 2016 *)
RecurrenceTable[{a[0]==1,a[1]==21,a[n]==20a[n-1]-64a[n-2]+2},a,{n,20}] (* Harvey P. Dale, Oct 27 2019 *)

Formula

a(n) = (257*16^n - 85*4^n + 8)/180.
G.f.: (1+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3) for n>2.
E.g.f.: (1/180)*(257*exp(16*x) - 85*exp(4*x) + 8*exp(x)). (End)

A167033 a(n) = 20*a(n-1) - 64*a(n-2) + 3 for n > 1; a(0) = 1, a(1) = 22.

Original entry on oeis.org

1, 22, 379, 6175, 99247, 1589743, 25443055, 407117551, 6513995503, 104224386799, 1667592023791, 26681479720687, 426903704891119, 6830459395698415, 109287350800936687, 1748597614694035183, 27977561842620755695
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A166915, A166916, A006105, A167031, A167032.

Programs

  • Magma
    [ n le 2 select 21*n-20 else 20*Self(n-1)-64*Self(n-2)+3: n in [1..17] ];
  • Mathematica
    LinearRecurrence[{21, -84, 64}, {1, 22, 379}, 50] (* G. C. Greubel, May 30 2016 *)

Formula

a(n) = (91*16^n - 35*4^n + 4)/60.
G.f.: (1+x+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/60)*(91*exp(16*x) - 35*exp(4*x) + 4*exp(x)). (End)
Showing 1-9 of 9 results.

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