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Triangle read by rows: binary products of Fibonacci numbers.
4

%I #13 Jan 05 2025 19:51:37

%S 1,2,3,5,6,8,13,15,16,21,34,39,40,42,55,89,102,104,105,110,144,233,

%T 267,272,273,275,288,377,610,699,712,714,715,720,754,987,1597,1830,

%U 1864,1869,1870,1872,1885,1974,2584,4181,4791,4880,4893,4895,4896,4901,4935,5168,6765

%N Triangle read by rows: binary products of Fibonacci numbers.

%C Row n consists of n numbers, first F(2n-1) and last F(2n).

%C Central numbers: (1,6,40,273,...) = A081016.

%C Row sums: A001870.

%C 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 In each row, the difference between neighboring terms is a Fibonacci number.

%H G. C. Greubel, <a href="/A094565/b094565.txt">Rows n = 1..100 of triangle, flattened</a>

%H 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 Row n: F(2)F(2n-1), F(4)F(2n-3), ..., F(2n)F(1).

%e Triangle begins:

%e 1;

%e 2, 3;

%e 5, 6 8;

%e 13, 15, 16, 21;

%e 34, 39, 40, 42, 55;

%e 89, 102, 104, 105, 110, 144; ...

%t Table[Fibonacci[2*k]*Fibonacci[2*n-2*k+1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 15 2019 *)

%o (PARI) row(n) = vector(n, k, fibonacci(2*k)*fibonacci(2*n-2*k+1));

%o tabl(nn) = for(n=1, nn, print(row(n))); \\ _Michel Marcus_, May 03 2016

%o (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 (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 (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 Cf. A000045, A094566, A094568.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, May 12 2004