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Triangle read by rows, X^n * [1,0,0,0,...]; where X = an infinite tridiagonal matrix with (1,0,1,0,1,...) in the main and subsubdiagonals and (1,1,1,...) in the subdiagonal.
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%I #11 Jun 09 2018 13:34:13

%S 1,1,1,1,1,1,3,1,1,1,1,5,3,5,1,1,1,1,7,5,13,5,7,1,1,1,1,9,7,25,13,25,

%T 7,9,1,1,1,1,11,9,41,25,63,25,41,9,11,1,1,1,1,13,11,61,41,129,63,129,

%U 41,61,11,13,1,1

%N Triangle read by rows, X^n * [1,0,0,0,...]; where X = an infinite tridiagonal matrix with (1,0,1,0,1,...) in the main and subsubdiagonals and (1,1,1,...) in the subdiagonal.

%C Row sums = A001333 starting (1, 3, 7, 17, 41, 99, 239,...).

%C Can also be seen as a triangle where each entry is the sum of two terms above it in previous row (as in Pascal's triangle) plus one term above it two rows back, see also A059317. - _Reinhard Zumkeller_, Jun 30 2012

%H Reinhard Zumkeller, <a href="/A140750/b140750.txt">Rows n = 1..100 of triangle, flattened</a>

%H S. Samieinia, <a href="http://dx.doi.org/10.4171/PM/1858">The number of continuous curves in digital geometry</a>, Port. Math. 67 (1) (2010) 75-89

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%e First few rows of the triangle are:

%e 1;

%e 1, 1, 1;

%e 1, 1, 3, 1, 1;

%e 1, 1, 5, 3, 5, 1, 1;

%e 1, 1, 7, 5, 13, 5, 7, 1, 1;

%e 1, 1, 9, 7, 25, 13, 25, 7, 9, 1, 1;

%e 1, 1, 11, 9, 41, 25, 63, 25, 41, 9, 11, 1, 1;

%e 1, 1, 13, 11, 61, 41, 129, 63, 129, 41, 61, 11, 13, 1, 1;

%e ...

%o (Haskell)

%o a140750 n k = a140750_tabf !! (n-1) !! (k-1)

%o a140750_row n = a140750_tabf !! (n-1)

%o a140750_tabf = [1] : [1,1,1] : f [1] [1,1,1] where

%o f ws vs = vs' : f vs vs' where

%o vs' = zipWith3 (\r s x -> r + s + x)

%o (vs ++ [0,0]) ([0,0] ++ ws ++ [0,0]) ([0,0] ++ vs)

%o -- _Reinhard Zumkeller_, Jun 30 2012

%Y Cf. A001333, A140751.

%Y Cf. A005408 (row lengths).

%K nonn,tabl

%O 1,7

%A _Gary W. Adamson_ & _Roger L. Bagula_, May 26 2008