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.

A164120 Partial sums of A162396.

Original entry on oeis.org

5, 7, 17, 21, 41, 49, 89, 105, 185, 217, 377, 441, 761, 889, 1529, 1785, 3065, 3577, 6137, 7161, 12281, 14329, 24569, 28665, 49145, 57337, 98297, 114681, 196601, 229369, 393209, 458745, 786425, 917497, 1572857, 1835001, 3145721, 3670009
Offset: 1

Views

Author

Klaus Brockhaus, Aug 10 2009

Keywords

Links

Crossrefs

Cf. A162396, A164053 (partial sums of A162255).

Programs

  • Magma
    T:=[ n le 2 select 8-3*n else 2*Self(n-2): n in [1..38] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];
  • Mathematica
    Rest[CoefficientList[Series[x*(5 + 2*x)/((1 - x)*(1 - 2*x^2)), {x,0,50}], x]] (* or *) LinearRecurrence[{1,2,-2}, {5,7,17}, 50] (* G. C. Greubel, Sep 12 2017 *)
  • PARI
    x='x+O('x^50); Vec(x*(5+2*x)/((1-x)*(1-2*x^2))) \\ G. C. Greubel, Sep 12 2017

Formula

a(n) = 2*a(n-2) + 7 for n > 2; a(1) = 5, a(2) = 7.
a(n) = (19 - 5*(-1)^n)*2^((2*n-1+(-1)^n)/4)/2 - 7.
G.f.: x*(5+2*x)/((1-x)*(1-2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3). - G. C. Greubel, Sep 12 2017

A162268 a(n) = ((5+sqrt(2))*(1+sqrt(2))^n + (5-sqrt(2))*(1-sqrt(2))^n)/2.

Original entry on oeis.org

5, 7, 19, 45, 109, 263, 635, 1533, 3701, 8935, 21571, 52077, 125725, 303527, 732779, 1769085, 4270949, 10310983, 24892915, 60096813, 145086541, 350269895, 845626331, 2041522557, 4928671445, 11898865447, 28726402339, 69351670125
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Binomial transform of A162396.

Links

Crossrefs

Cf. A162396.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(1+r)^n+(5-r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Mathematica
    LinearRecurrence[{2,1}, {5,7}, 30] (* Vincenzo Librandi, Feb 03 2018 *)
Table[(4*LucasL[n, 2] + LucasL[n + 1, 2])/2, {n, 0, 30}] (* G. C. Greubel, Aug 17 2018 *)
  • PARI
    x='x+O('x^30); Vec((5-3*x)/(1-2*x-x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 7.
G.f.: (5-3*x)/(1-2*x-x^2).
a(n) = 5*A000129(n+1) - 3*A000129(n). - R. J. Mathar, Mar 06 2013
a(n) = 4*A001333(n) + A001333(n+1). - G. C. Greubel, Aug 17 2018

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009

A162269 a(n) = ((5+sqrt(2))*(2+sqrt(2))^n + (5-sqrt(2))*(2-sqrt(2))^n)/2.

Original entry on oeis.org

5, 12, 38, 128, 436, 1488, 5080, 17344, 59216, 202176, 690272, 2356736, 8046400, 27472128, 93795712, 320238592, 1093362944, 3732974592, 12745172480, 43514740736, 148568617984, 507244990464, 1731842725888, 5912880922624
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Second binomial transform of A162396.

Links

Crossrefs

Cf. A162396.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(2+r)^n+(5-r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Mathematica
    LinearRecurrence[{4,-2},{5,12},30] (* Harvey P. Dale, Jan 03 2016 *)
  • PARI
    x='x+O('x^30); Vec((5-8*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Oct 02 2018

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 5, a(1) = 12.
G.f.: (5-8*x)/(1-4*x+2*x^2).
From G. C. Greubel, Oct 02 2018: (Start)
a(2*n) = 2^(n-1)*(Q(2*n+1) + 4*Q(2*n)), where Q(m) = Pell-Lucas numbers.
a(2*n+1) = 2^(n+1)*(P(2*n+2) + 4*P(2*n+1)), where P(m) = Pell numbers. (End)
a(n) = 5*A007070(n)-8*A007070(n-1). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009

A162270 a(n) = ((5+sqrt(2))*(3+sqrt(2))^n + (5-sqrt(2))*(3-sqrt(2))^n)/2.

Original entry on oeis.org

5, 17, 67, 283, 1229, 5393, 23755, 104779, 462389, 2040881, 9008563, 39765211, 175531325, 774831473, 3420269563, 15097797067, 66644895461, 294184793297, 1298594491555, 5732273396251, 25303478936621, 111694959845969
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Third binomial transform of A162396.

Links

Crossrefs

Cf. A162396.

Programs

  • GAP
    a:=[5,17];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018
  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(3+r)^n+(5-r)*(3-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Magma
    I:=[5,17]; [n le 2 select I[n] else 6*Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018
  • Maple
    seq(coeff(series((5-13*x)/(1-6*x+7*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018
  • Mathematica
    LinearRecurrence[{6,-7},{5,17},30] (* Harvey P. Dale, Jun 04 2016 *)
  • PARI
    x='x+O('x^30); Vec((5-13*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 28 2018

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 17.
G.f.: (5-13*x)/(1-6*x+7*x^2).
a(n) = 5*A081179(n+1)-13*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009

A162271 a(n) = ((5+sqrt(2))*(4+sqrt(2))^n + (5-sqrt(2))*(4-sqrt(2))^n)/2.

Original entry on oeis.org

5, 22, 106, 540, 2836, 15128, 81320, 438768, 2371664, 12830560, 69441184, 375901632, 2035036480, 11017668992, 59650841216, 322959363840, 1748563133696, 9467073975808, 51256707934720, 277514627816448
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Fourth binomial transform of A162396.

Links

Crossrefs

Cf. A162396.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(4+r)^n+(5-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Mathematica
    LinearRecurrence[{8,-14}, {5,22}, 50] (* G. C. Greubel, Oct 02 2018 *)
Table[((5+Sqrt[2])(4+Sqrt[2])^n+(5-Sqrt[2])(4-Sqrt[2])^n)/2,{n,0,20}]// Simplify (* Harvey P. Dale, May 26 2019 *)
  • PARI
    x='x+O('x^50); Vec((5-18*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Oct 02 2018

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 22.
G.f.: (5-18*x)/(1-8*x+14*x^2).

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.