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A094566 Triangle of binary products of Fibonacci numbers.

%I A094566 #11 Jan 05 2025 19:51:37
%S A094566 1,1,2,3,4,5,8,9,10,13,21,24,25,26,34,55,63,64,65,68,89,144,165,168,
%T A094566 169,170,178,233,377,432,440,441,442,445,466,610,987,1131,1152,1155,
%U A094566 1156,1157,1165,1220,1597,2584,2961,3016,3024,3025,3026,3029,3050,3194,4181
%N A094566 Triangle of binary products of Fibonacci numbers.
%C A094566 For n>1, row n consists of n numbers, first F(2n-2) and last F(2n-1).
%C A094566 Central numbers: (1,4,9,25,64,...), essentially A081016.
%C A094566 Row sums: A027991. Alternating row sums: 1,1,4,4,30,30,203,203; the sequence b=(1,4,30,203,1394,...) is A094567.
%C A094566 In each row, the difference between neighboring terms is a Fibonacci number.
%H A094566 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 A094566 Row 1 is the single number 1. For m>=1, Row 2m: F(2)F(4m-2), F(4)F(4m-4), ..., F(2m)F(2m), F(2m+1)F(2m-1), F(2m+3)F(2m-3), ..., F(4m-1)F(1) Row 2m+1: F(2)F(4m), F(4)F(4m-2), ..., F(2m+1)F(2m+1), F(2m+3)F(2m-1), F(2m+5)F(2m-3), ..., F(4m+1)F(1)
%e A094566 Rows 1 to 4:
%e A094566 1
%e A094566 1 2
%e A094566 3 4 5
%e A094566 8 9 10 13
%o A094566 (PARI) pef(k, n) = fibonacci(2*k)*fibonacci(2*n-2*k);
%o A094566 pof(k, n) = fibonacci(2*n-2*k+1)*fibonacci(2*k-1);
%o A094566 tabl(nn) = {for (n=1, nn, if (n==1, print1(1, ", "), if (n % 2 == 0, for (k=1, n/2, print1(pef(k,n), ", ");); forstep (k=n/2, 1, -1, print1(pof(k,n), ", "););, for (k=1, n\2, print1(pef(k,n), ", ");); forstep (k=n\2+1, 1, -1, print1(pof(k,n), ", ");););); print(););} \\ _Michel Marcus_, May 04 2016
%Y A094566 Cf. A000045, A094565, A094567, A094568.
%K A094566 nonn,tabl
%O A094566 1,3
%A A094566 _Clark Kimberling_, May 12 2004