0,3

Using the continued fraction method, given the denominators of

[1,N,1,N,1,N,...], where the N's begin (0,1,2,3,...).

If the first denominator is indexed "1", then extract the even-indexed denominators.

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,.......................................

...

Example: (3, 11, 41,....) is extracted from denominators of the continued

fraction [1, 2, 1, 2, 1, 2,...] = [1, 3, 4, 11, 15, 41,...].

Using the matrix method, (4, 19, 91,...) can be extracted as terms (1,1)

and (2,2) in powers of the matrices.

Row sums of the traingle = A162998: (1, 3, 29, 100, 369, 1458,...).

Columns of the array (>0) have trigonometric properties relating to the odd

N-gons as product formulas; such that (1, 5, 11, 19, 29,...) relates to

the Pentagon, (1, 13, 41, 91,...) relates to the Heptagon, and so on:

If we relabel columns m=(3, 5, 7,...) and rows r=(2, 3, 4,...) then for

columns (>3), the array term = PRODUCT_{k=1..(N-1)/2} (r + 2*Cos 2Pi/m).

Example: the term "41" in the relabeled array would be r=4, m=7, so

41 = (5.24697,...) * (3.554,...) * (2.19806,...).

Descending diagonals of an array generated from 2x2 matrices of the form

[1,N; 1,(N+1)]; then extracting alternate terms. Alternatively, given continued fractions of the form [1,N,1,N,1,N,...] extract alternate terms of the denominators.

First few rows of the triangle =

1;

1, 2;

1, 5, 3;

1, 13, 11, 4;

1, 34, 41, 19, 5;

1, 89, 153, 91, 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;

... Q

A162988, A028387

Sequence in context: A125171 A048472 A038622 this_sequence A112339 A132808 A135233

Adjacent sequences: A162994 A162995 A162996 this_sequence A162998 A162999 A163000

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2009