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Triangular array read by rows: T(n,k) is the quotient on division of Fib(n) by Fib(k) for 1 <= k <= n, where Fib(k) = A000045(k).
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%I #33 Oct 25 2022 20:04:07

%S 1,1,1,2,2,1,3,3,1,1,5,5,2,1,1,8,8,4,2,1,1,13,13,6,4,2,1,1,21,21,10,7,

%T 4,2,1,1,34,34,17,11,6,4,2,1,1,55,55,27,18,11,6,4,2,1,1,89,89,44,29,

%U 17,11,6,4,2,1,1,144,144,72,48,28,18,11,6,4,2,1,1,233,233,116,77,46,29,17,11

%N Triangular array read by rows: T(n,k) is the quotient on division of Fib(n) by Fib(k) for 1 <= k <= n, where Fib(k) = A000045(k).

%H Robert Israel, <a href="/A357814/b357814.txt">Table of n, a(n) for n = 1..10011</a>(rows 1 to 141, flattened)

%F T(n,k) = floor(A000045(n)/A000045(k)).

%F A000045(n) = T(n,k)*A000045(k) + A357724(n,k).

%F The following follow from the identity A000045(n) = A000045(k)*A000032(n-k) + (-1)^k*A000045(n-2*k).

%F T(2*m*k,k) = Sum_{j=0..m-1} (-1)^((j+m+1)*k)*A000032((2*j+1)*k).

%F T((2*m+1)*k,k) = -1 + Sum_{j=0..m} (-1)^((j+m)*k)*A000032(2*j*k) if k is even.

%F T((2*m+1)*k,k) = (-1)^(m+1) + Sum_{j=0..m} (-1)^((j+m)*k)*A000032(2*j*k) if k is odd.

%F If 2*m*k < n < (2*m+1)*k and k >= 4 is even, then T(n,k) = Sum_{j=0..m-1} A000032(n-(2*j+1)*k)

%F If 2*m*k < n < (2*m+1)*k, m is even and k is odd, then T(n,k) = Sum_{j=0..m-1} (-1)^j*A000032(n-(2*j+1)*k).

%F If 2*m*k < n < (2*m+1)*k, m is odd and k is odd, then T(n,k) = -1 + Sum_{j=0..m-1} (-1)^j*A000032(n-(2*j+1)*k).

%F If (2*m+1)*k < n < (2*m+2)*k, and either k is odd and n+m is even, or k >= 4 is even and n is odd, then T(n,k) = Sum_{j=0..m} (-1)^j*A000032(n-(2*j+1)*k).

%F If (2*m+1)*k < n < (2*m+2)*k, and either k is odd and n+m is odd, or k >= 4 is even and n is even, then T(n,k) = -1 + Sum_{j=0..m} (-1)^j*A000032(n-(2*j+1)*k).

%e Triangle starts:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 3, 3, 1, 1;

%e 5, 5, 2, 1, 1;

%e 8, 8, 4, 2, 1, 1;

%e 13, 13, 6, 4, 2, 1, 1;

%e 21, 21, 10, 7, 4, 2, 1, 1;

%e 34, 34, 17, 11, 6, 4, 2, 1, 1;

%e 55, 55, 27, 18, 11, 6, 4, 2, 1, 1;

%e 89, 89, 44, 29, 17, 11, 6, 4, 2, 1, 1;

%p f:= (n,k) -> iquo(combinat:-fibonacci(n), combinat:-fibonacci(k)):

%p for n from 1 to 12 do

%p seq(f(n,k),k=1..n)

%p od:

%Y Cf. A000045, A000032, A357724.

%K nonn,tabl,look

%O 1,4

%A _J. M. Bergot_ and _Robert Israel_, Oct 13 2022