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Ordered differences of Fibonacci numbers.
32

%I #38 Aug 27 2024 02:55:18

%S 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,

%T 13,54,53,52,50,47,42,34,21,88,87,86,84,81,76,68,55,34,143,142,141,

%U 139,136,131,123,110,89,55,232,231,230,228,225,220,212,199,178

%N Ordered differences of Fibonacci numbers.

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

%C From _Emanuele Munarini_, Mar 29 2012: (Start)

%C Diagonal elements = Fibonacci numbers F(n+1) (A000045)

%C First column = Fibonacci numbers - 1 (A000071);

%C Second column = Fibonacci numbers - 2 (A001911);

%C Row sums = n*F(n+3) - F(n+2) + 2 (A014286);

%C Central coefficients = F(2*n+1) - F(n+1) (A096140).

%C (End)

%H G. C. Greubel, <a href="/A204922/b204922.txt">Rows n=1..100 of triangle, flattened</a>

%F From _Emanuele Munarini_, Mar 29 2012: (Start)

%F T(n,k) = Fibonacci(n+2) - Fibonacci(k+1).

%F T(n,k) = Sum_{i=k..n} Fibonacci(i+1). (End)

%e a(1) = s(2) - s(1) = F(3) - F(2) = 2-1 = 1, where F=A000045;

%e a(2) = s(3) - s(1) = F(4) - F(2) = 3-1 = 2;

%e a(3) = s(3) - s(2) = F(4) - F(3) = 3-2 = 1;

%e a(4) = s(4) - s(1) = F(5) - F(2) = 5-1 = 4.

%e From _Emanuele Munarini_, Mar 29 2012: (Start)

%e Triangle begins:

%e 1;

%e 2, 1;

%e 4, 3, 2;

%e 7, 6, 5, 3;

%e 12, 11, 10, 8, 5;

%e 20, 19, 18, 16, 13, 8;

%e 33, 32, 31, 29, 26, 21, 13;

%e 54, 53, 52, 50, 47, 42, 34, 21;

%e 88, 87, 86, 84, 81, 76, 68, 55, 34;

%e ... (End)

%t (See the program at A204924.)

%o (Maxima) create_list(fib(n+3)-fib(k+2),n,0,20,k,0,n); /* _Emanuele Munarini_ */

%o (Magma) /* As triangle */ [[Fibonacci(n+2)-Fibonacci(k+1) : k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Aug 04 2015

%o (PARI) {T(n,k) = fibonacci(n+2) - fibonacci(k+1)};

%o for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 03 2019

%o (Sage) [[fibonacci(n+2) - fibonacci(k+1) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Feb 03 2019

%Y Cf. A204924, A204892.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 21 2012