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A143211 Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).

%I A143211 #13 Oct 02 2024 04:18:03
%S A143211 1,1,1,2,2,4,3,3,6,9,5,5,10,15,25,8,8,16,24,40,64,13,13,26,39,65,104,
%T A143211 169,21,21,42,63,105,168,273,441,34,34,68,102,170,272,442,714,1156,55,
%U A143211 55,110,165,275,440,715,1155,1870,3025,89,89,178,267,445,712,1157,1869
%N A143211 Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).
%H A143211 G. C. Greubel, <a href="/A143211/b143211.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A143211 T(n, k) = Fibonacci(n)*Fibonacci(k).
%F A143211 T(n, k) = A127647 * A000012 * A127647, as infinite lower triangular matrices.
%F A143211 T(n, 1) = A000045(n).
%F A143211 T(n, n) = A007598(n).
%F A143211 Sum_{k=1..n} T(n, k) = A143212(n).
%F A143211 From _G. C. Greubel_, Jul 20 2024: (Start)
%F A143211 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*Fibonacci(n)*(Fibonacci(n-1) - (-1)^n).
%F A143211 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024458(n). (End)
%e A143211 First few rows of the triangle:
%e A143211    1;
%e A143211    1,  1;
%e A143211    2,  2,  4;
%e A143211    3,  3,  6,  9;
%e A143211    5,  5, 10, 15,  25;
%e A143211    8,  8, 16, 24,  40,  64;
%e A143211   13, 13, 26, 39,  65, 104, 169;
%e A143211   21, 21, 42, 63, 105, 168, 273, 441;
%e A143211   ...
%t A143211 With[{F=Fibonacci}, Table[F[k]*F[n], {n,12}, {k,n}]]//Flatten (* _G. C. Greubel_, Jul 20 2024 *)
%o A143211 (Magma) F:=Fibonacci; [F(n)*F(k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 20 2024
%o A143211 (SageMath)
%o A143211 def A143211(n,k): return fibonacci(n)*fibonacci(k)
%o A143211 flatten([[A143211(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jul 20 2024
%Y A143211 Cf. A000045 (left border), A007598 (right border), A127647,
%Y A143211 Cf. A024458 (diagonal row sums), A143212 (row sums).
%K A143211 nonn,tabl,easy
%O A143211 1,4
%A A143211 _Gary W. Adamson_, Jul 30 2008