cp's OEIS Frontend

A162997 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.

%I A162997 #25 Jul 25 2024 04:41:25
%S A162997 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 A162997 169,41,7,1,610,2131,2089,985,281,55,8,1,1597,7953,10009,5741,1926,
%U A162997 433,71,9
%N A162997 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 A162997 With k=0 column added, becomes A094954.
%C A162997 Also, A(n,k) is the top-left element of the same 2 X 2 matrix raised to (k+1)-th power.
%C A162997 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 A162997 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 A162997 Row sums of the triangle = A162998: (1, 3, 9, 29, 100, 369, 1458, ...).
%H A162997 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 A162997 The array begins:
%e A162997 1,...1,...1,....1,....1,.....1,.....1,...
%e A162997 2,...5,..13,...34,...89,...233....610,...
%e A162997 3,..11,..41,..153,..571,..2131,..........
%e A162997 4,..19,..91,..436,.2089,.................
%e A162997 5,..29,.169,..985,.......................
%e A162997 6,..41,.281,.............................
%e A162997 7,..55,..................................
%e A162997 8,.......................................
%e A162997 ...
%Y A162997 Cf. A028387, A094954, A162998, A152063.
%K A162997 nonn,tabl
%O A162997 0,3
%A A162997 _Gary W. Adamson_, Jul 19 2009
%E A162997 Spelling corrected by _Jason G. Wurtzel_, Aug 22 2010
%E A162997 Edited by _Andrey Zabolotskiy_, Sep 18 2017