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 1-3 of 3 results.

A014286 a(n) = Sum_{j=0..n} j*Fibonacci(j).

Original entry on oeis.org

0, 1, 3, 9, 21, 46, 94, 185, 353, 659, 1209, 2188, 3916, 6945, 12223, 21373, 37165, 64314, 110826, 190265, 325565, 555431, 945073, 1604184, 2717016, 4592641, 7748859, 13052145, 21950853, 36863494, 61824694, 103559033, 173264921, 289575995, 483474153
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Equals row sums of triangle A143061. - Gary W. Adamson, Jul 20 2008

Links

Crossrefs

Cf. A000045.
Cf. A143061.
Partial sums of A045925.
Cf. A282464: partial sums of j*Fibonacci(j)^2.

Programs

  • GAP
    List([0..50], n-> n*Fibonacci(n+2)-Fibonacci(n+3)+2); # G. C. Greubel, Jun 13 2019
  • Magma
    [n*Fibonacci(n+2)-Fibonacci(n+3)+2: n in [0..50]]; // Vincenzo Librandi, Mar 31 2011
  • Maple
    A014286 := proc(n)
    add(i*combinat[fibonacci](i),i=0..n) ;
end proc: # R. J. Mathar, Apr 11 2016
  • Mathematica
    Accumulate[Table[Fibonacci[n]*n, {n, 0, 50}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 9; a[n_] := a[n] = 2 a[n-1] + a[n-2] - 2 a[n-3] - a[n-4] + 2; Table[a[n], {n, 0, 50}] (* Vladimir Reshetnikov, Oct 28 2015 *)
  • PARI
    concat(0, Vec(x*(1+x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Oct 28 2015
  • Sage
    [n*fibonacci(n+2)-fibonacci(n+3)+2 for n in (0..50)] # G. C. Greubel, Jun 13 2019

Formula

G.f.: x*(1+x^2)/((1-x)*(1-x-x^2)^2).
a(n) = n*F(n+2) - F(n+3) + 2.
Recurrences, from Vladimir Reshetnikov, Oct 28 2015: (Start)
6-term, homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(4) = 21, a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).
5-term, non-homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + 2. (End)

A171729 Triangle of differences of Fibonacci numbers, rows ascending.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 3, 4, 5, 3, 5, 6, 7, 8, 5, 8, 10, 11, 12, 13, 8, 13, 16, 18, 19, 20, 21, 13, 21, 26, 29, 31, 32, 33, 34, 21, 34, 42, 47, 50, 52, 53, 54, 55, 34, 55, 68, 76, 81, 84, 86, 87, 88, 89, 55, 89, 110, 123, 131, 136, 139, 141, 142, 143, 144, 89, 144, 178, 199, 212, 220, 225, 228, 230, 231, 232, 233
Offset: 1

Views

Author

Clark Kimberling, Dec 16 2009

Keywords

Comments

The numbers missing from this triangle form A050939.
Row n of this triangle has one more term than row n of A143061.
Reversing the rows gives A171730.

Examples

			First rows:
  1
  1 2
  1 2  3
  2 3  4  5
  3 5  6  7  8
  5 8 10 11 12 13
  ...

Links

Crossrefs

Cf. A000045, A050939, A143061, A171730.

Programs

  • Maple
    F:= combinat[fibonacci]:
T:= (n,k)-> F(n+1)-`if`(k=n, 0, F(n-k+1)):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 06 2023
  • Mathematica
    Table[Fibonacci[n + 1] - If[k < n, Fibonacci[n - k + 1], 0], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Feb 06 2023 *)
  • PARI
    row(n) = vector(n, k, fibonacci(n+1) - if (kMichel Marcus, Feb 06 2023

Formula

Counting the top row as the first row, the n-th row is
F(n+1)-F(n), F(n+1)-F(n-1), ..., F(n+1)-F(2), F(n+1)-F(0).

A171730 Triangle of differences of Fibonacci numbers, rows descending.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 2, 8, 7, 6, 5, 3, 13, 12, 11, 10, 8, 5, 21, 20, 19, 18, 16, 13, 8, 34, 33, 32, 31, 29, 26, 21, 13, 55, 54, 53, 52, 50, 47, 42, 34, 21, 89, 88, 87, 86, 84, 81, 76, 68, 55, 34, 144, 143, 142, 141, 139, 136, 131, 123, 110, 89, 55, 233, 232, 231, 230, 228, 225, 220, 212, 199, 178, 144, 89
Offset: 1

Views

Author

Clark Kimberling, Dec 16 2009

Keywords

Comments

The numbers missing from this triangle form A050939.
Reversing the rows gives A171729.

Examples

			First rows:
   1
   2  1
   3  2  1
   5  4  3  2
   8  7  6  5 3
  13 12 11 10 8 5
  ...

Links

Crossrefs

Cf. A000045, A050939, A143061, A171729.

Programs

  • Maple
    F:= combinat[fibonacci]:
T:= (n,k)-> F(n+1)-`if`(k=1, 0, F(k)):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 06 2023
  • Mathematica
    Table[Fibonacci[n + 1] - If[k > 1, Fibonacci[k], 0], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Feb 06 2023 *)
  • PARI
    row(n) = vector(n, k, fibonacci(n+1) - if (k>1, fibonacci(k), 0)); \\ Michel Marcus, Feb 06 2023

Formula

Counting the top row as the first row, the n-th row is
F(n+1)-F(0), F(n+1)-F(2), ..., F(n+1)-F(n-1), F(n+1)-F(n).
Showing 1-3 of 3 results.

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

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