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%I #34 Sep 08 2022 08:46:01
%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 Triangle begins:
%C 1;
%C 2, 1;
%C 4, 3, 2;
%C 7, 6, 5, 3;
%C 12, 11, 10, 8, 5;
%C 20, 19, 18, 16, 13, 8;
%C 33, 32, 31, 29, 26, 21, 13;
%C 54, 53, 52, 50, 47, 42, 34, 21;
%C 88, 87, 86, 84, 81, 76, 68, 55, 34;
%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.
%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
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