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 1-4 of 4 results.

A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

Original entry on oeis.org

3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

Comments

Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.

Links

Crossrefs

Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009
  • Magma
    [Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
  • Maple
    with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011
  • Mathematica
    Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* Alonso del Arte, Nov 29 2010 *)
LinearRecurrence[{4,1},{3,11},30] (* Harvey P. Dale, Apr 14 2021 *)
  • PARI
    Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

Formula

a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
G.f.: (3-x)/(1-4*x-x^2).
a(n) = A033887(n) + A014445(n+1).
a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - Wolfdieter Lang, Mar 09 2011.
a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Dec 10 2015

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163062 a(n) = ((3+sqrt(5))*(1+sqrt(5))^n + (3-sqrt(5))*(1-sqrt(5))^n)/2.

Original entry on oeis.org

3, 8, 28, 88, 288, 928, 3008, 9728, 31488, 101888, 329728, 1067008, 3452928, 11173888, 36159488, 117014528, 378667008, 1225392128, 3965452288, 12832473088, 41526755328, 134383403008, 434873827328, 1407281266688
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

Comments

Binomial transform of A163114. Inverse binomial transform of A163063.

Links

Crossrefs

Cf. A000032, A163063, A163114.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(1+r)^n+(3-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009
  • Magma
    I:=[3,8]; [n le 2 select I[n] else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017
  • Mathematica
    CoefficientList[Series[(3+2*x)/(1-2*x-4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,4}, {3,8}, 30] (* G. C. Greubel, Dec 22 2017 *)
  • PARI
    x='x+O('x^30); Vec((3+2*x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 3, a(1) = 8.
G.f.: (3+2*x)/(1-2*x-4*x^2).
a(n) = 2^n * A000032(n+2). - Diego Rattaggi, Jun 17 2020

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163064 a(n) = ((3+sqrt(5))*(4+sqrt(5))^n + (3-sqrt(5))*(4-sqrt(5))^n)/2.

Original entry on oeis.org

3, 17, 103, 637, 3963, 24697, 153983, 960197, 5987763, 37339937, 232854103, 1452093517, 9055353003, 56469795337, 352149479663, 2196028088597, 13694580432483, 85400334485297, 532562291125063, 3321094649662237
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

Comments

Binomial transform of A098648 without initial 1. Fourth binomial transform of A163114. Inverse binomial transform of A163065.

Links

Crossrefs

Cf. A098648, A163114, A163065.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009
  • Magma
    I:=[3,17]; [n le 2 select I[n] else 8*Self(n-1) - 11*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017
  • Mathematica
    CoefficientList[Series[(3-7*x)/(1-8*x+11*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{8,-11}, {3,17}, 30] (* G. C. Greubel, Dec 22 2017 *)
  • PARI
    x='x+O('x^30); Vec((3-7*x)/(1-8*x+11*x^2)) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 3, a(1) = 17.
G.f.: (3-7*x)/(1-8*x+11*x^2).

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163065 a(n) = ((3+sqrt(5))*(5+sqrt(5))^n + (3-sqrt(5))*(5-sqrt(5))^n)/2.

Original entry on oeis.org

3, 20, 140, 1000, 7200, 52000, 376000, 2720000, 19680000, 142400000, 1030400000, 7456000000, 53952000000, 390400000000, 2824960000000, 20441600000000, 147916800000000, 1070336000000000, 7745024000000000, 56043520000000000
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

Comments

Binomial transform of A163064. Fifth binomial transform of A163114.
10^(floor(n/2)) | a(n). - G. C. Greubel, Dec 22 2017

Links

Crossrefs

Cf. A163064, A163114.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009
  • Magma
    I:=[3,20]; [n le 2 select I[n] else 10*Self(n-1) - 20*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017
  • Mathematica
    CoefficientList[Series[(3-10*x)/(1-10*x+20*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{10,-20}, {3,20}, 30] (* G. C. Greubel, Dec 22 2017 *)
  • PARI
    x='x+O('x^30); Vec((3-10*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = 10*a(n-1) - 20*a(n-2) for n > 1; a(0) = 3, a(1) = 20.
G.f.: (3-10*x)/(1-10*x+20*x^2).

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009
Showing 1-4 of 4 results.

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