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.

A204922 Ordered differences of Fibonacci numbers.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892. For numbers not in A204922, see A050939.
From Emanuele Munarini, Mar 29 2012: (Start)
Diagonal elements = Fibonacci numbers F(n+1) (A000045)
First column = Fibonacci numbers - 1 (A000071);
Second column = Fibonacci numbers - 2 (A001911);
Row sums = n*F(n+3) - F(n+2) + 2 (A014286);
Central coefficients = F(2*n+1) - F(n+1) (A096140).
(End)

Examples

			a(1) = s(2) - s(1) = F(3) - F(2) = 2-1 = 1, where F=A000045;
a(2) = s(3) - s(1) = F(4) - F(2) = 3-1 = 2;
a(3) = s(3) - s(2) = F(4) - F(3) = 3-2 = 1;
a(4) = s(4) - s(1) = F(5) - F(2) = 5-1 = 4.
From _Emanuele Munarini_, Mar 29 2012: (Start)
Triangle begins:
   1;
   2,  1;
   4,  3,  2;
   7,  6,  5,  3;
  12, 11, 10,  8,  5;
  20, 19, 18, 16, 13,  8;
  33, 32, 31, 29, 26, 21, 13;
  54, 53, 52, 50, 47, 42, 34, 21;
  88, 87, 86, 84, 81, 76, 68, 55, 34;
  ... (End)

Links

Crossrefs

Cf. A204924, A204892.

Programs

  • Magma
    /* As triangle */ [[Fibonacci(n+2)-Fibonacci(k+1) : k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 04 2015
  • Mathematica
    (See the program at A204924.)
  • Maxima
    create_list(fib(n+3)-fib(k+2),n,0,20,k,0,n); /* Emanuele Munarini, Mar 29 2012 */
  • PARI
    {T(n,k) = fibonacci(n+2) - fibonacci(k+1)};
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 03 2019
  • Sage
    [[fibonacci(n+2) - fibonacci(k+1) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Feb 03 2019

Formula

From Emanuele Munarini, Mar 29 2012: (Start)
T(n,k) = Fibonacci(n+2) - Fibonacci(k+1).
T(n,k) = Sum_{i=k..n} Fibonacci(i+1). (End)

A007298 Sums of consecutive Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 18, 19, 20, 21, 26, 29, 31, 32, 33, 34, 42, 47, 50, 52, 53, 54, 55, 68, 76, 81, 84, 86, 87, 88, 89, 110, 123, 131, 136, 139, 141, 142, 143, 144, 178, 199, 212, 220, 225, 228, 230, 231, 232, 233, 288, 322
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2000

Keywords

Comments

Also the differences between two Fibonacci numbers, because the difference F(i+2) - F(j+1) equals the sum F(j) + ... + F(i). - T. D. Noe, Oct 17 2005; corrected by Patrick Capelle, Mar 01 2008

Links

Crossrefs

Cf. A000045, A050939.
Cf. A113188 (primes that are the difference of two Fibonacci numbers).
Cf. A219114 (numbers whose squares are here).

Programs

  • Maple
    isA007298 := proc(n)
    local i,Fi,j,Fj ;
    for i from 0 do
        Fi := combinat[fibonacci](i) ;
        for j from i do
            Fj :=combinat[fibonacci](j) ;
            if Fj-Fi = n then
                return true;
            elif Fj-Fi > n then
                break;
            end if;
        end do:
        Fj :=combinat[fibonacci](i+1) ;
        if Fj-Fi > n then
            return false;
        end if;
    end do:
end proc:
for n from 0 to 100 do
    if isA007298(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016
  • Mathematica
    Union[Flatten[Table[Fibonacci[n]-Fibonacci[i], {n, 14}, {i, n}]]] (* T. D. Noe, Oct 17 2005 *)
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[0, 1000], isA007298] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar *)
  • PARI
    A130233(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2)
list(lim)=my(v=List([0]),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); listput(v,s))); Set(v) \\ Charles R Greathouse IV, Oct 06 2016

Formula

log a(n) >> sqrt(n). - Charles R Greathouse IV, Oct 06 2016

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 1-4 of 4 results.

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

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