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 1-6 of 6 results.

A059509 Main diagonal of the array A059503.

Original entry on oeis.org

1, 5, 19, 66, 216, 679, 2075, 6211, 18299, 53244, 153366, 438095, 1242709, 3504161, 9830371, 27454614, 76375860, 211732471, 585157679, 1612689439, 4433421131, 12160156560, 33284285874, 90931830431, 247991356201, 675243561149, 1835863145395, 4984516006506, 13516071450384
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Links

Crossrefs

Cf. A059503.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(x^3 - x + 1)/(x^2 - 3*x + 1)^2, {x,0,50}], x]] (* or *) Table[((3 - n)*Fibonacci[2*n] - (5 - 8*n)*Fibonacci[2*n - 1])/5, {n, 1, 50}] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(x^3-x+1)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(x^3-x+1)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) - (5 - 8*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017
E.g.f.: 1 + exp(3*x/2)*(5*(7*x - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x + 11)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Apr 11 2025

A059502 a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, 43906, 122868, 342409, 950727, 2631165, 7260579, 19982612, 54865566, 150316973, 411015705, 1121818311, 3056773383, 8316416134, 22593883752, 61301547025, 166118284299, 449639574897, 1215751720491, 3283883157848
Offset: 0

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence.
Variation of A059216 (and of Boustrophedon transform) applied to 1,2,3,4,...: fill an array by diagonals, each time in the same direction, say the 'up' direction. The first column is 1,2,3,4,... For the next element of a diagonal, add to the previous element the elements of the row the new element is in. The first row gives a(n).

Examples

			The array (see A059503) begins
  1 3  9 27 80 ...
  2 5 14 40 ...
  3 7 19 ...
  4 9  5 ...

Links

Crossrefs

Cf. A000045, A000667, A059216, A059219, A027994, A059512, A059503.

Programs

  • Magma
    [(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // Vincenzo Librandi, Apr 23 2011
  • Mathematica
    Table[(3n Fibonacci[2n-1]+(3-n)Fibonacci[2n])/5,{n,0,30}] (* or *) CoefficientList[Series[x(1-x)(1-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* Harvey P. Dale, Apr 23 2011 *)
  • PARI
    a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5

Formula

a(n) = 2*a(n-1) + Sum{m<=n-2} a(m) + A001519(n-2).
G.f.: x*(1 - x)*(1 - 2*x)/(1 - 3*x + x^2)^2. - Emeric Deutsch, Oct 07 2002
a(n) = A147703(n,1). - Philippe Deléham, Nov 29 2008
a(n) = A001871(n-1) - 3*A001871(n-2) + 2*A001871(n-3). - R. J. Mathar, Apr 09 2019
E.g.f.: 2*exp(3*x/2)*(5*x*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

A059505 Transform of A059502 applied to sequence 2,3,4,...

Original entry on oeis.org

2, 5, 14, 40, 114, 323, 910, 2551, 7120, 19796, 54852, 151525, 417434, 1147145, 3145394, 8606848, 23507190, 64093031, 174474790, 474261691, 1287398452, 3490267820, 9451319304, 25565098825, 69080289074
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The second row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{2,5,14,40}, 50] (* or *) Rest[CoefficientList[Series[x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    x='x+O('x^50); Vec(x*(2-7*x+6*x^2-x^3)/(1-3*x+x^2)^2) \\ G. C. Greubel, Sep 10 2017

Formula

G.f.: x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2.
From G. C. Greubel, Sep 10 2017: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
a(n) = ((3 - n)*Fibonacci(2*n) + (5 + 3*n)*Fibonacci(2*n - 1))/5. (End)

A059506 Transform of A059502 applied to sequence 3,4,5,...

Original entry on oeis.org

3, 7, 19, 53, 148, 412, 1143, 3161, 8717, 23977, 65798, 180182, 492459, 1343563, 3659623, 9953117, 27031768, 73320496, 198632607, 537507677, 1452978593, 3923762257, 10586222474, 28536313898, 76859031123
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The third row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{3,7,19,53},30] (* Harvey P. Dale, Jul 30 2015 *)
Rest[CoefficientList[Series[x*(1 - x)*(2*x^2 - 8*x + 3)/(x^2 - 3*x + 1)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2 + O(x^30)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (10 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059507 Transform of A059502 applied to sequence 4,5,6,...

Original entry on oeis.org

4, 9, 24, 66, 182, 501, 1376, 3771, 10314, 28158, 76744, 208839, 567484, 1539981, 4173852, 11299386, 30556346, 82547961, 222790424, 600753663, 1618558734, 4357256694, 11721125644, 31507528971, 84637773172
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fourth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1 - x)*(3*x^2 - 11*x + 4)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (15 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059508 Transform of A059502 applied to sequence 5,6,7,...

Original entry on oeis.org

5, 11, 29, 79, 216, 590, 1609, 4381, 11911, 32339, 87690, 237496, 642509, 1736399, 4688081, 12645655, 34080924, 91775426, 246948241, 663999649, 1784138875, 4790751131, 12856028814, 34478744044, 92416515221
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fifth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502, A059503.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1-x)*(4*x^2 - 14*x + 5)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(-x*(x-1)*(4*x^2-14*x+5)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(4*x^2-14*x+5)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (20 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017
Showing 1-6 of 6 results.

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

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