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A102310 Square array read by antidiagonals: Fibonacci(k*n).

%I A102310 #22 Sep 08 2022 08:45:16
%S A102310 1,1,1,2,3,2,3,8,8,3,5,21,34,21,5,8,55,144,144,55,8,13,144,610,987,
%T A102310 610,144,13,21,377,2584,6765,6765,2584,377,21,34,987,10946,46368,
%U A102310 75025,46368,10946,987,34,55,2584,46368,317811,832040,832040,317811,46368,2584,55
%N A102310 Square array read by antidiagonals: Fibonacci(k*n).
%D A102310 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994, p. 294.
%H A102310 Freddy Barrera, <a href="/A102310/b102310.txt">Antidiagonals n = 1..50, flattened</a>
%F A102310 For prime p, the formula holds: Fibonacci(k*p) = Fibonacci(p) * Sum_{i=0..floor((k-1)/2)} C(k-i-1, i)*(-1)^(i*p+i)*Lucas(p)^(k-2i-1).
%F A102310 A(n, k) = F((n-1)*k)*F(k+1) + F((n-1)*k-1)*F(k), where F(n) = A000045(n). - _Freddy Barrera_, Jun 24 2019
%e A102310 1,  1,   2,    3,     5, ...
%e A102310 1,  3,   8,   21,    55, ...
%e A102310 2,  8,  34,  144,   610, ...
%e A102310 3, 21, 144,  987,  6765, ...
%e A102310 5, 55, 610, 6765, 75025, ...
%t A102310 Table[Fibonacci[k*(n-k+1)], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2017 *)
%o A102310 (Sage)
%o A102310 F = fibonacci # A000045
%o A102310 def A(n, k):
%o A102310     return F((n-1)*k)*F(k+1) + F((n-1)*k - 1)*F(k)
%o A102310 [A(n, k) for d in (1..10) for n, k in zip((d..1, step=-1), (1..d))] # _Freddy Barrera_, Jun 24 2019
%o A102310 (Magma) /* As triangle */ [[Fibonacci(k*(n-k+1)): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Jul 04 2019
%Y A102310 Equals A000045(A003991(k, n)).
%Y A102310 Columns include A000045, A001906, A014445, A033888, A102312.
%Y A102310 Main diagonal is in A054783. Antidiagonal sums are in A102311.
%K A102310 nonn,tabl
%O A102310 1,4
%A A102310 _Ralf Stephan_, Jan 06 2005