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.

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

Offset

0,3

Comments

With k=0 column added, becomes A094954.

Also, A(n,k) is the top-left 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, 9, 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, 22-31.

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, A162998, 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