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.

Previous Showing 11-15 of 15 results.

A157729 a(n) = Fibonacci(n) + 5.

Original entry on oeis.org

5, 6, 6, 7, 8, 10, 13, 18, 26, 39, 60, 94, 149, 238, 382, 615, 992, 1602, 2589, 4186, 6770, 10951, 17716, 28662, 46373, 75030, 121398, 196423, 317816, 514234, 832045, 1346274, 2178314, 3524583, 5702892, 9227470, 14930357, 24157822, 39088174, 63245991, 102334160
Offset: 0

Views

Author

N. J. A. Sloane, Jun 26 2010

Keywords

Links

Crossrefs

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616.

Programs

  • Haskell
    a157729 = (+ 5) . a000045
a157729_list = 5 : 6 : map (subtract 5)
                       (zipWith (+) a157729_list $ tail a157729_list)
-- Reinhard Zumkeller, Jul 30 2013
  • Magma
    [ Fibonacci(n) + 5: n in [0..40] ]; // Vincenzo Librandi, Apr 24 2011
  • Mathematica
    Fibonacci[Range[0,40]]+5 (* or *) LinearRecurrence[{2,0,-1},{5,6,6},50] (* Harvey P. Dale, Aug 17 2012 *)
  • PARI
    a(n)=fibonacci(n)+5 \\ Charles R Greathouse IV, Jul 02 2013

Formula

G.f.: ( 5-4*x-6*x^2 ) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Aug 09 2012
a(0)=5, a(1)=6, a(2)=6, a(n)=2*a(n-1)+0*a(n-2)-a(n-3). - Harvey P. Dale, Aug 17 2012
a(0) = 5, a(1) = 6, a(n) = a(n - 2) + a(n - 1) - 5. - Reinhard Zumkeller, Jul 30 2013

A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

Original entry on oeis.org

0, 5, 13, 26, 47, 81, 136, 225, 369, 602, 979, 1589, 2576, 4173, 6757, 10938, 17703, 28649, 46360, 75017, 121385, 196410, 317803, 514221, 832032, 1346261, 2178301, 3524570, 5702879, 9227457, 14930344, 24157809, 39088161, 63245978, 102334147, 165580133
Offset: 0

Views

Author

Johannes W. Meijer, Sep 21 2010

Keywords

Comments

The a(n+1) (terms doubled) are the Kn15 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Links

Crossrefs

Cf. A000045.
Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

Programs

  • GAP
    List([0..40], n-> Fibonacci(n+6)-8); # G. C. Greubel, Jul 13 2019
  • Magma
    [Fibonacci(n+6)-Fibonacci(6): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011
  • Maple
    nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+6)-fibonacci(6) od: seq(a(n),n=0..nmax);
  • Mathematica
    f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)
LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)
CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* Robert G. Wilson v, Apr 11 2017 *)
  • PARI
    for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ Anton Mosunov, Mar 02 2017
  • PARI
    concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Apr 20 2017
  • Sage
    [fibonacci(n+6)-8 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+6) - F(6) with F = A000045.
a(n) = a(n-1) + a(n-2) + 8 for n>1, a(0)=0, a(1)=5, and where 8 = F(6).
From Colin Barker, Apr 13 2012: (Start)
G.f.: x*(5 + 3*x)/((1 - x)*(1 - x - x^2)).
a(n) = 2*a(n-1) - a(n-3). (End)
a(n) = (-8 + (2^(-n)*((1-sqrt(5))^n*(-9+4*sqrt(5)) + (1+sqrt(5))^n*(9+4*sqrt(5)))) / sqrt(5)). - Colin Barker, Apr 20 2017

A180672 a(n) = Fibonacci(n+7) - Fibonacci(7).

Original entry on oeis.org

0, 8, 21, 42, 76, 131, 220, 364, 597, 974, 1584, 2571, 4168, 6752, 10933, 17698, 28644, 46355, 75012, 121380, 196405, 317798, 514216, 832027, 1346256, 2178296, 3524565, 5702874, 9227452, 14930339, 24157804, 39088156, 63245973
Offset: 0

Views

Author

Johannes W. Meijer, Sep 21 2010

Keywords

Comments

The a(n+1) (terms doubled) are the Kn16 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Links

Crossrefs

Cf. A000045, A000071.
Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

Programs

  • GAP
    List([0..40], n-> Fibonacci(n+7)-13 ); # G. C. Greubel, Jul 13 2019
  • Magma
    [Fibonacci(n+7) - Fibonacci(7): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011
  • Maple
    nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+7)-fibonacci(7) od: seq(a(n),n=0..nmax);
  • Mathematica
    Fibonacci[7 +Range[0, 40]] -13 (* G. C. Greubel, Jul 13 2019 *)
  • PARI
    concat(0, Vec(x*(8+5*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017
  • PARI
    a(n)=fibonacci(n+7)-fibonacci(7) \\ Charles R Greathouse IV, Feb 24 2017
  • Sage
    [fibonacci(n+7)-13 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+7) - F(7) with F = A000045.
a(n) = a(n-1) + a(n-2) + 13 for n>1, a(0)=0, a(1)=8, and where 13 = F(7).
G.f.: x*(8 + 5*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-13 + (2^(-1-n)*((1-sqrt(5))^n*(-29+13*sqrt(5)) + (1+sqrt(5))^n*(29+13*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 8*A000071(n+2) + 5*A000071(n+1). - Bruno Berselli, Feb 24 2017

A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

Original entry on oeis.org

0, 13, 34, 68, 123, 212, 356, 589, 966, 1576, 2563, 4160, 6744, 10925, 17690, 28636, 46347, 75004, 121372, 196397, 317790, 514208, 832019, 1346248, 2178288, 3524557, 5702866, 9227444, 14930331, 24157796, 39088148, 63245965, 102334134
Offset: 0

Views

Author

Johannes W. Meijer, Sep 21 2010

Keywords

Comments

The a(n+1) (terms doubled) are the Kn17 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Links

Crossrefs

Cf. A000045, A000071.
Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

Programs

  • GAP
    List([0..40], n-> Fibonacci(n+8)-21); # G. C. Greubel, Jul 13 2019
  • Magma
    [Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011
  • Maple
    nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+8)-fibonacci(8) od: seq(a(n),n=0..nmax);
  • Mathematica
    Fibonacci[8 +Range[0, 40]] -21 (* G. C. Greubel, Jul 13 2019 *)
  • PARI
    concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017
  • PARI
    a(n)=fibonacci(n+8)-21 \\ Charles R Greathouse IV, Feb 24 2017
  • SageMath
    [fibonacci(n+8)-21 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+8) - F(8) with F(n) the Fibonacci numbers A000045.
a(n) = a(n-1) + a(n-2) + 21 for n>1, a(0)=0, a(1)=13, and where 21 = F(8).
G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 13*A000071(n+2) + 8*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-21 + (2^(-1-n)*((1-sqrt(5))^n*(-47+21*sqrt(5)) + (1+sqrt(5))^n*(47+21*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A180674 a(n) = Fibonacci(n+9) - Fibonacci(9).

Original entry on oeis.org

0, 21, 55, 110, 199, 343, 576, 953, 1563, 2550, 4147, 6731, 10912, 17677, 28623, 46334, 74991, 121359, 196384, 317777, 514195, 832006, 1346235, 2178275, 3524544, 5702853, 9227431, 14930318, 24157783, 39088135, 63245952, 102334121
Offset: 0

Views

Author

Johannes W. Meijer, Sep 21 2010

Keywords

Comments

The a(n+1) (terms doubled) are the Kn18 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Links

Crossrefs

Cf. A000045, A000071.
Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

Programs

  • GAP
    List([0..40], n-> Fibonacci(n+9)-34); # G. C. Greubel, Jul 13 2019
  • Magma
    [Fibonacci(n+9) - Fibonacci(9): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011
  • Maple
    nmax:=31: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+9)-fibonacci(9) od: seq(a(n),n=0..nmax);
  • Mathematica
    Fibonacci[9 +Range[0, 40]] -34 (* G. C. Greubel, Jul 13 2019 *)
LinearRecurrence[{2,0,-1},{0,21,55},40] (* Harvey P. Dale, Aug 24 2024 *)
  • PARI
    concat(0, Vec(x*(21+13*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017
  • PARI
    a(n) = fibonacci(n+9) - fibonacci(9) \\ Charles R Greathouse IV, Feb 24 2017
  • Sage
    [fibonacci(n+9)-34 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+9) - F(9) with F = A000045.
a(n) = a(n-1) + a(n-2) + 34 for n>1, a(0)=0, a(1)=21, and where 34 = F(9).
G.f.: x*(21 + 13*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 21*A000071(n+2) + 13*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-34 + (2^(-n)*((1-sqrt(5))^n*(-38+17*sqrt(5)) + (1+sqrt(5))^n*(38+17*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
Previous Showing 11-15 of 15 results.

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

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