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

Author

Clark Kimberling, Jan 21 2012

Keywords

• nonn
• tabl

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

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

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)