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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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, 29, 100, 369, 1458, ...).
LINKS
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
Sequence in context: A048472 A038622 A193954 * A112339 A132808 A135233
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

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Last modified April 16 11:08 EDT 2024. Contains 371711 sequences. (Running on oeis4.)