This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194582 #42 Feb 28 2020 13:25:00
%S A194582 1,0,1,0,3,1,0,2,6,1,0,2,13,9,1,0,2,16,33,12,1,0,2,20,69,62,15,1,0,2,
%T A194582 24,108,188,100,18,1,0,2,28,156,401,400,147,21,1,0,2,32,212,704,1115,
%U A194582 732,203,24,1,0,2,36,276,1120,2433,2547,1211,268,27,1
%N A194582 Triangle T(n,k), read by rows, given by (0, 3, -7/3, -2/21, 3/7, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A194582 Riordan array (1, x*(1+2x-x^2)/(1-x)).
%C A194582 Row sums are (Fibonacci(n+1))^2 = A007598(n+1).
%C A194582 T(n, k) is the number of ordered pairs of Fibonacci bit strings of length n with the number of matching 1 bits in the same position is k. A Fibonacci bit string begins a 1 bit and no two consecutive bits are 0 bits. - _Michael Somos_, Feb 28 2020
%H A194582 Indranil Ghosh, <a href="/A194582/b194582.txt">Rows 0..100, flattened</a>
%F A194582 T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1) - T(n-3,k-1).
%F A194582 G.f.: (1-x)/(1-x-x*y-2*x^2*y+x^3*y). - _R. J. Mathar_, Aug 11 2015
%e A194582 Triangle begins:
%e A194582 1;
%e A194582 0, 1;
%e A194582 0, 3, 1;
%e A194582 0, 2, 6, 1;
%e A194582 0, 2, 13, 9, 1;
%e A194582 0, 2, 16, 33, 12, 1;
%e A194582 0, 2, 20, 69, 62, 15, 1;
%e A194582 0, 2, 24, 108, 188, 100, 18, 1;
%e A194582 0, 2, 28, 156, 401, 400, 147, 21, 1;
%e A194582 T(3, 2) = 6 enumerates the pairs of Fibonacci bit string of length 3 with 2 matching 1 bits: (101, 101), (101, 111), (110, 110), (110, 111), (111, 101), (111, 110). - _Michael Somos_, Feb 28 2020
%t A194582 nmax=10; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - x)/(1 - x - x*y - 2*x^2*y + x^3*y) , {x, 0, nmax}], x], {y, 0, nmax}], y]] (* _Indranil Ghosh_, Mar 10 2017, after _R. J. Mathar_ *)
%o A194582 (PARI) T(n,k) = if(n==k, 1, if(k==0, 0, if(n>1 && k==n - 1, 3*k, T(n - 1, k) + T(n - 1,k - 1) + 2*T(n - 2,k - 1) - T(n-3,k-1))));
%o A194582 {for(n=0, 10, for(k=0, n, print1(T(n,k),", ");); print();); } \\ _Indranil Ghosh_, Mar 10 2017
%Y A194582 Cf. A000045, A007598. Diagonals: A000012, A008585, A062708.
%K A194582 nonn,tabl
%O A194582 0,5
%A A194582 _Philippe Deléham_, Jan 23 2012