

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

Table of n, a(n) for n=0..44.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 2231.


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

Cf. A028387, A094954, A162988, A152063.
Sequence in context: A048472 A038622 A193954 * A112339 A132808 A135233
Adjacent sequences: A162994 A162995 A162996 * A162998 A162999 A163000


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Jul 19 2009


EXTENSIONS

Spelling corrected by Jason G. Wurtzel, Aug 22 2010
Edited by Andrey Zabolotskiy, Sep 18 2017


STATUS

approved



