%I #15 Jul 25 2021 03:36:01

%S 10,1001,1010,100101,101001,101010,10010101,10100101,10101001,

%T 10101010,1001010101,1010010101,1010100101,1010101001,1010101010,

%U 100101010101,101001010101,101010010101,101010100101,101010101001,101010101010

%N Triangle read by rows of numbers with n 1's and n 0's in their representation in base of Fibonacci numbers (A210619), written as those 1's and 0's.

%C The digits of T(n,k) are k pairs 10 followed by n-k pairs 01.

%H Kevin Ryde, <a href="/A346434/b346434.txt">Table of n, a(n) for triangle rows 1 to 49, flattened</a>

%F T(n,k) = (10*100^n - 9*100^(n-k) - 1)/99, for n>=1 and 1 <= k <= n.

%F T(n,k) = A014417(A210619(n,k)).

%F T(n,n) = A163662(n).

%F G.f.: x*y*(10 - 9*x - 100*x^2*y) / ((1-x) * (1-100*x) * (1-x*y) * (1-100*x*y) ).

%e Triangle begins:

%e k=1 k=2 k=3 k=4

%e n=1: 10

%e n=2: 1001, 1010,

%e n=3: 100101, 101001, 101010,

%e n=4: 10010101, 10100101, 10101001, 10101010

%e ...

%e For n=5,k=3, the 10 and 01 digit pairs are

%e vvvvvv k = 3 pairs 10

%e T(5,3) = 1010100101

%e ^^^^ n-k = 2 pairs 01

%o (PARI) T(n,k) = (10*100^n - 9*100^(n-k)) \ 99;

%Y Cf. A210619, A163662 (main diagonal), A014417 (Zeckendorf digits).

%K nonn,easy,tabl

%O 1,1

%A _Kevin Ryde_, Jul 18 2021