A204922 Ordered differences of Fibonacci numbers.

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

1,2

Comments

For a guide to related sequences, see A204892. For numbers not in A204922, see A050939.

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;

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)

Links

G. C. Greubel, Rows n=1..100 of triangle, flattened

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)

Example

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.

Mathematica

(See the program at A204924.)

Prog

(Maxima) create_list(fib(n+3)-fib(k+2), n, 0, 20, k, 0, n); /* Emanuele Munarini */
(Magma) /* As triangle */ [[Fibonacci(n+2)-Fibonacci(k+1) : k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 04 2015
(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

Crossrefs

Cf. A204924, A204892.

Sequence in context: A287010 A144330 A141155 * A057669 A243610 A182013

Adjacent sequences: A204919 A204920 A204921 * A204923 A204924 A204925

Keyword

nonn

Author

Clark Kimberling, Jan 21 2012

Status

approved