

A162997


Array A(n,k) read by antidiagonals downward (n >= 0, k >= 1): the bottomright element of the 2 X 2 matrix [1,n; 1,n+1] raised to kth power.


6



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, 169, 41, 7, 1, 610, 2131, 2089, 985, 281, 55, 8, 1, 1597, 7953, 10009, 5741, 1926, 433, 71, 9
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OFFSET

0,3


COMMENTS

With k=0 column added, becomes A094954.
Also, A(n,k) is the topleft element of the same 2 X 2 matrix raised to (k+1)th power.
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, ...
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.
Row sums of the triangle = A162998: (1, 3, 29, 100, 369, 1458, ...).


LINKS



EXAMPLE

The array begins:
1,...1,...1,....1,....1,.....1,.....1,...
2,...5,..13,...34,...89,...233....610,...
3,..11,..41,..153,..571,..2131,..........
4,..19,..91,..436,.2089,.................
5,..29,.169,..985,.......................
6,..41,.281,.............................
7,..55,..................................
8,.......................................
...


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



