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A094565 Triangle read by rows: binary products of Fibonacci numbers.

%I A094565 #13 Jan 05 2025 19:51:37
%S A094565 1,2,3,5,6,8,13,15,16,21,34,39,40,42,55,89,102,104,105,110,144,233,
%T A094565 267,272,273,275,288,377,610,699,712,714,715,720,754,987,1597,1830,
%U A094565 1864,1869,1870,1872,1885,1974,2584,4181,4791,4880,4893,4895,4896,4901,4935,5168,6765
%N A094565 Triangle read by rows: binary products of Fibonacci numbers.
%C A094565 Row n consists of n numbers, first F(2n-1) and last F(2n).
%C A094565 Central numbers: (1,6,40,273,...) = A081016.
%C A094565 Row sums: A001870.
%C A094565 Alternating row sums: 1,1,7,7,48,48,329,329; the sequence b=(1,7,48,329,...) is A004187, given by b(n)=F(4n+2)-b(n-1) for n>=2, with b(1)=1.
%C A094565 In each row, the difference between neighboring terms is a Fibonacci number.
%H A094565 G. C. Greubel, <a href="/A094565/b094565.txt">Rows n = 1..100 of triangle, flattened</a>
%H A094565 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-1/quartkimberling01_2004.pdf">Orderings of products of Fibonacci numbers</a>, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
%F A094565 Row n: F(2)F(2n-1), F(4)F(2n-3), ..., F(2n)F(1).
%e A094565 Triangle begins:
%e A094565    1;
%e A094565    2,   3;
%e A094565    5,   6    8;
%e A094565   13,  15,  16,  21;
%e A094565   34,  39,  40,  42,  55;
%e A094565   89, 102, 104, 105, 110, 144; ...
%t A094565 Table[Fibonacci[2*k]*Fibonacci[2*n-2*k+1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 15 2019 *)
%o A094565 (PARI) row(n) = vector(n, k, fibonacci(2*k)*fibonacci(2*n-2*k+1));
%o A094565 tabl(nn) = for(n=1, nn, print(row(n))); \\ _Michel Marcus_, May 03 2016
%o A094565 (Magma) [Fibonacci(2*k)*Fibonacci(2*n-2*k+1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 15 2019
%o A094565 (Sage) [[fibonacci(2*k)*fibonacci(2*n-2*k+1) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 15 2019
%o A094565 (GAP) Flat(List([1..12], n-> List([1..n], k-> Fibonacci(2*k)*Fibonacci(2*n-2*k+1) ))); # _G. C. Greubel_, Jul 15 2019
%Y A094565 Cf. A000045, A094566, A094568.
%K A094565 nonn,tabl
%O A094565 1,2
%A A094565 _Clark Kimberling_, May 12 2004