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%I #21 Sep 19 2017 05:40:35
%S 1,1,2,1,5,3,1,13,11,4,1,34,41,19,5,1,89,153,92,29,6,1,233,571,436,
%T 169,41,7,1,610,2131,2089,985,281,55,8,1,1597,7953,10009,5741,1926,
%U 433,71,9
%N Array A(n,k) read by antidiagonals downward (n >= 0, k >= 1): the bottom-right element of the 2 X 2 matrix [1,n; 1,n+1] raised to k-th power.
%C With k=0 column added, becomes A094954.
%C Also, A(n,k) is the top-left element of the same 2 X 2 matrix raised to (k+1)-th power.
%C Also, A(n,k) is the denominator of the rational number which has continued fraction expansion consisting of k repeats of [1, n]. Example: the row (3, 11, 41, ...) is extracted from denominators of the continued fractions [0; 1, 2], [0; 1, 2, 1, 2], ... = 2/3, 8/11, ...
%C Also, A(n,k)=Product_{i=1..k} (n+2+2*cos(2*Pi*i/(2*k+1))). This is somehow connected to the diagonal product formulas for (2*k+1)-gons found by Steinbach.
%C Row sums of the triangle = A162998: (1, 3, 29, 100, 369, 1458, ...).
%H P. Steinbach, <a href="https://doi.org/10.2307/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%e The array begins:
%e 1,...1,...1,....1,....1,.....1,.....1,...
%e 2,...5,..13,...34,...89,...233....610,...
%e 3,..11,..41,..153,..571,..2131,..........
%e 4,..19,..91,..436,.2089,.................
%e 5,..29,.169,..985,.......................
%e 6,..41,.281,.............................
%e 7,..55,..................................
%e 8,.......................................
%e ...
%Y Cf. A028387, A094954, A162988, A152063.
%K nonn,tabl
%O 0,3
%A _Gary W. Adamson_, Jul 19 2009
%E Spelling corrected by _Jason G. Wurtzel_, Aug 22 2010
%E Edited by _Andrey Zabolotskiy_, Sep 18 2017
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