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Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).
4

%I #13 Oct 02 2024 04:18:03

%S 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 169,21,21,42,63,105,168,273,441,34,34,68,102,170,272,442,714,1156,55,

%U 55,110,165,275,440,715,1155,1870,3025,89,89,178,267,445,712,1157,1869

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

%H G. C. Greubel, <a href="/A143211/b143211.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = Fibonacci(n)*Fibonacci(k).

%F T(n, k) = A127647 * A000012 * A127647, as infinite lower triangular matrices.

%F T(n, 1) = A000045(n).

%F T(n, n) = A007598(n).

%F Sum_{k=1..n} T(n, k) = A143212(n).

%F From _G. C. Greubel_, Jul 20 2024: (Start)

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*Fibonacci(n)*(Fibonacci(n-1) - (-1)^n).

%F Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024458(n). (End)

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 2, 2, 4;

%e 3, 3, 6, 9;

%e 5, 5, 10, 15, 25;

%e 8, 8, 16, 24, 40, 64;

%e 13, 13, 26, 39, 65, 104, 169;

%e 21, 21, 42, 63, 105, 168, 273, 441;

%e ...

%t With[{F=Fibonacci}, Table[F[k]*F[n], {n,12}, {k,n}]]//Flatten (* _G. C. Greubel_, Jul 20 2024 *)

%o (Magma) F:=Fibonacci; [F(n)*F(k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 20 2024

%o (SageMath)

%o def A143211(n,k): return fibonacci(n)*fibonacci(k)

%o flatten([[A143211(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jul 20 2024

%Y Cf. A000045 (left border), A007598 (right border), A127647,

%Y Cf. A024458 (diagonal row sums), A143212 (row sums).

%K nonn,tabl,easy

%O 1,4

%A _Gary W. Adamson_, Jul 30 2008