cp's OEIS Frontend

A341616 Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...).

%I A341616 #24 Mar 28 2021 00:22:29
%S A341616 1,3,3,8,4,4,21,14,7,7,55,33,22,11,11,144,90,54,36,18,18,377,232,145,
%T A341616 87,58,29,29,987,611,376,235,141,94,47,47,2584,1596,988,608,380,228,
%U A341616 152,76,76,6765,4182,2583,1599,984,615,369,246,123,123
%N A341616 Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...).
%C A341616 j is the offset when combining terms from the two initial sequences.
%F A341616 For phi=(1+sqrt(5))/2 and tau=(1-sqrt(5))/2:
%F A341616 T(n,j) = Fibonacci(n)*Lucas(n+j).
%F A341616 T(n,j) = (phi^n - tau^n)*(phi^(n+j) + tau^(n+j))/sqrt(5).
%F A341616 T(n,j) = Fibonacci(2n+j) - (-1)^n*Fibonacci(j).
%F A341616 Lim_{n, j -> oo} T(n+1,j)/T(n,j) = phi^2 (A104457).
%F A341616 Lim_{n, j -> oo} T(n,j+1)/T(n,j) = phi (A001622).
%e A341616 T(4,3) = Fibonacci(4)*Lucas(4+3) = 3*29 = 87.
%e A341616 Square array showing T(n,j) begins:
%e A341616       j=0 j=1 j=2 j=3 j=4  ..
%e A341616   n=1   1   3   4   7  11  ..
%e A341616   n=2   3   4   7  11  18  ..
%e A341616   n=3   8  14  22  36  58  ..
%e A341616   n=4  21  33  54  87 141  ..
%e A341616   ...  ..  ..  ..  ..  ..  ..
%o A341616 (PARI) T(n,j) = fibonacci(2*n+j) - (-1)^n*fibonacci(j);
%o A341616 matrix(7,7,n,k, T(n,k-1)) \\ _Michel Marcus_, Mar 02 2021
%Y A341616 For j=0 the resulting sequence is used as input in A341414.
%Y A341616 Cf. A000045, A000032, A001622, A094214, A104457.
%K A341616 easy,nonn,tabl
%O A341616 1,2
%A A341616 _Jens Rasmussen_, Feb 16 2021