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 1-5 of 5 results.

A072261 a(n) = 4*a(n-1) + 1, a(1)=7.

Original entry on oeis.org

7, 29, 117, 469, 1877, 7509, 30037, 120149, 480597, 1922389, 7689557, 30758229, 123032917, 492131669, 1968526677, 7874106709, 31496426837, 125985707349, 503942829397, 2015771317589, 8063085270357, 32252341081429, 129009364325717, 516037457302869
Offset: 1

Views

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

Comments

These are the integers N which on application of the Collatz function yield the number 11. The Collatz function: if N is an odd number then (3N+1)/2^r yields a positive odd integer for some value of r (which in this case is 11).
These numbers reach 11 in Collatz function iteration after 2(n+1) steps and so end in 1 after exactly 2n+18 steps. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004
Numbers whose binary representation is 111 together with n - 1 times 01. For example, a(4) = 469 = 111010101 (2). - Omar E. Pol, Nov 24 2012

Links

Crossrefs

Cf. A002450, A072197, A072257, A072258, A072259, A072260, A072262, A099730, A178415, A347834.

Programs

  • GAP
    List([1..25], n-> (11*4^n -2)/6); # G. C. Greubel, Jan 14 2020
  • Magma
    [(11*4^n -2)/6: n in [1..25]]; // G. C. Greubel, Jan 14 2020
  • Maple
    seq(coeff(series(x*(7-6*x)/((1-x)*(1-4*x)), x, n+1), x, n), n = 1..25); # G. C. Greubel, Jan 14 2020
  • Mathematica
    a[n_]:= 4a[n-1] +1; a[1]=7; Table[a[n], {n,25}]
NestList[4#+1&,7,30] (* or *) LinearRecurrence[{5,-4},{7,29},30] (* Harvey P. Dale, Sep 04 2023 *)
  • PARI
    Vec(x*(7-6*x)/((1-x)*(1-4*x)) + O(x^25)) \\ Colin Barker, Oct 27 2019
  • Sage
    [(11*4^n -2)/6 for n in (1..25)] # G. C. Greubel, Jan 14 2020

Formula

a(n) = (11*4^n - 2)/6 = 22*A002450(n-1) + 7. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004
From Colin Barker, Oct 27 2019: (Start)
G.f.: x*(7 - 6*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n>2. (End)
E.g.f.: (-9 - 2*exp(x) + 11*exp(4*x))/6. - G. C. Greubel, Jan 14 2020
a(n) = a(n-1) + 11*2^(2*n-3), for n >= 2, with a(1) = 7. - Wolfdieter Lang, Aug 16 2021
a(n) = A178415(4, n) = A347834(3, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

Extensions

Edited and extended by Robert G. Wilson v, Jul 17 2002
More terms from Colin Barker, Oct 27 2019

A072257 a(n) = ((6*n-17)*4^n - 1)/3.

Original entry on oeis.org

-6, -15, -27, 21, 597, 4437, 25941, 136533, 677205, 3233109, 15029589, 68506965, 307582293, 1364546901, 5995058517, 26127717717, 113100805461, 486762960213, 2084490794325, 8887718991189
Offset: 0

Views

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

Comments

Related to Collatz function (for n>2). All terms are divisible by 3.

Links

Crossrefs

Cf. A072258, A072259, A072260.

Programs

  • GAP
    List([0..40], n-> ((6*n-17)*4^n -1)/3); # G. C. Greubel, Jan 14 2020
  • Magma
    [((6*n-17)*4^n -1)/3: n in [0..40]]; // G. C. Greubel, Jan 14 2020
  • Maple
    seq( ((6*n-17)*4^n -1)/3, n=0..40); # G. C. Greubel, Jan 14 2020
  • Mathematica
    LinearRecurrence[{9,-24,16},{-6,-15,-27},40] (* Harvey P. Dale, Nov 23 2015 *)
  • PARI
    a(n)=((6*n-17)*4^n-1)/3 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [((6*n-17)*4^n -1)/3 for n in (0..40)] # G. C. Greubel, Jan 14 2020

Formula

G.f.: -3*(2-13*x+12*x^2)/((1-x)*(1-4*x)^2). - Bruno Berselli, Dec 16 2011
E.g.f.: (-1/3)*( (17-24*x)*exp(4*x) + exp(x) ). - G. C. Greubel, Jan 14 2020

Extensions

Edited and extended by Henry Bottomley, Aug 06 2002

A072259 a(n) = ((6*n+37)*4^n - 1)/3.

Original entry on oeis.org

12, 57, 261, 1173, 5205, 22869, 99669, 431445, 1856853, 7951701, 33903957, 144004437, 609572181, 2572506453, 10826896725, 45455070549, 190410216789, 796000605525, 3321441375573, 13835521316181, 57541108520277, 238960527103317, 991026480502101
Offset: 0

Views

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

Comments

Related to Collatz function (for n>0). All terms are divisible by 3.

Links

Crossrefs

Cf. A072257, A072258, A072260.

Programs

  • GAP
    List([0..30], n-> ((6*n+37)*4^n -1)/3); # G. C. Greubel, Jan 14 2020
  • Magma
    [((6*n+37)*4^n -1)/3: n in [0..30]]; // G. C. Greubel, Jan 14 2020
  • Maple
    seq( ((6*n+37)*4^n -1)/3, n=0..30); # G. C. Greubel, Jan 14 2020
  • Mathematica
    LinearRecurrence[{9,-24,16},{12,57,261},30] (* Harvey P. Dale, Mar 10 2018 *)
  • PARI
    a(n)=((6*n+37)*4^n-1)/3 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [((6*n+37)*4^n -1)/3 for n in (0..30)] # G. C. Greubel, Jan 14 2020

Formula

G.f.: 3*(4-17*x+12*x^2)/((1-x)*(1-4*x)^2). - Bruno Berselli, Dec 16 2011
E.g.f.: ((37 + 24*x)*exp(4*x) - exp(x))/3. - G. C. Greubel, Jan 14 2020

Extensions

Edited and extended by Henry Bottomley, Aug 06 2002
More terms from Harvey P. Dale, Mar 10 2018

A072260 a(n) = ((6*n+19)*4^n - 1)/3.

Original entry on oeis.org

6, 33, 165, 789, 3669, 16725, 75093, 333141, 1463637, 6378837, 27612501, 118838613, 508908885, 2169853269, 9216283989, 39012619605, 164640413013, 692921390421, 2909124515157
Offset: 0

Views

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

Comments

Related to Collatz function (for n>0). All terms are divisible by 3.

Links

Crossrefs

Cf. A072257, A072258, A072259.

Programs

  • GAP
    List([0..20], n-> ((6*n+19)*4^n-1)/3); # G. C. Greubel, Jan 14 2020
  • Magma
    [((6*n+19)*4^n-1)/3: n in [0..20]]; // G. C. Greubel, Jan 14 2020
  • Maple
    seq( ((6*n+19)*4^n -1)/3, n=0..20); # G. C. Greubel, Jan 14 2020
  • Mathematica
    LinearRecurrence[{9,-24,16}, {6,33,165}, 20] (* G. C. Greubel, Jan 14 2020 *)
Table[((6n+19)4^n-1)/3,{n,0,20}] (* Harvey P. Dale, Jun 20 2024 *)
  • PARI
    a(n)=((6*n+19)*4^n-1)/3 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [((6*n+19)*4^n-1)/3 for n in (0..20)] # G. C. Greubel, Jan 14 2020

Formula

G.f.: 3*(2-7*x+4*x^2)/((1-x)*(1-4*x)^2). - Bruno Berselli, Dec 16 2011
E.g.f.: ((19 + 24*x)*exp(4*x) - exp(x))/3. - G. C. Greubel, Jan 14 2020

Extensions

Edited and extended by Henry Bottomley, Aug 06 2002

A072262 a(n) = 4*a(n-1) + 1, a(1)=11.

Original entry on oeis.org

11, 45, 181, 725, 2901, 11605, 46421, 185685, 742741, 2970965, 11883861, 47535445, 190141781, 760567125, 3042268501, 12169074005, 48676296021, 194705184085, 778820736341, 3115282945365, 12461131781461, 49844527125845
Offset: 1

Views

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

Comments

These are the integers N which on application of the Collatz function yield the number 17. The Collatz function: if N is an odd number then (3N+1)/2^r yields a positive odd integer for some value of r (which in this case is 17).
Numbers whose binary representation is 1011 together with n - 1 times 01. For example, a(4) = 725 = 1011010101 (2). - Omar E. Pol, Nov 24 2012

Links

Crossrefs

Cf. A072257, A072258, A072259, A072260, A072261, A099730, A178415, A347834.

Programs

  • GAP
    List([1..30], n-> (17*4^n -2)/6); # G. C. Greubel, Jan 14 2020
  • Magma
    [(17*4^n -2)/6: n in [1..30]]; // G. C. Greubel, Jan 14 2020
  • Maple
    seq( (17*4^n -2)/6, n=1..30); # G. C. Greubel, Jan 14 2020
  • Mathematica
    a[n_]:= 4a[n-1] +1; a[1]=11; Table[a[n], {n,25}]
NestList[4#+1&,11,30] (* or *) LinearRecurrence[{5,-4},{11,45},30] (* Harvey P. Dale, Dec 25 2014 *)
  • PARI
    vector(30, n, (17*4^n -2)/6) \\ G. C. Greubel, Jan 14 2020
  • Sage
    [(17*4^n -2)/6 for n in (1..30)] # G. C. Greubel, Jan 14 2020

Formula

From Bruno Berselli, Dec 16 2011: (Start)
G.f.: x*(11-10*x)/(1-5*x+4*x^2).
a(n) = (17*2^(2*n-1) - 1)/3.
Sum_{i=1..n} a(i) = (a(n+1) - n + 1)/3 - 4. (End)
a(n) = 34*A002450(n-1) + 11 . - Yosu Yurramendi, Jan 24 2017
E.g.f.: (-15 - 2*exp(x) + 17*exp(4*x))/6. - G. C. Greubel, Jan 14 2020
a(n) = A178415(6, n) = A347834(5, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

Extensions

Edited and extended by Robert G. Wilson v, Jul 17 2002
Showing 1-5 of 5 results.

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