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A102541 Triangle read by rows, formed from antidiagonals of Losanitsch's triangle. T(n,k ) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2). 34
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 3, 6, 2, 1, 4, 9, 6, 1, 1, 4, 12, 10, 3, 1, 5, 16, 19, 9, 1, 1, 5, 20, 28, 19, 3, 1, 6, 25, 44, 38, 12, 1, 1, 6, 30, 60, 66, 28, 4, 1, 7, 36, 85, 110, 66, 16, 1, 1, 7, 42, 110, 170, 126, 44, 4, 1, 8, 49, 146, 255, 236, 110, 20, 1, 1, 8, 56 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums A102526 are essentially the same as A001224, A060312 and A068928.

Moving the terms in each column of this triangle, see the example, upwards to row 0 gives  Losanitsch’s triangle A034851 as a square array. - Johannes W. Meijer, Aug 24 2013

The number of ways to cover n-length line by exactly k 2-length segments excluding symmetric covers. - Philipp O. Tsvetkov, Nov 08 2013

Also the number of equivalence classes of ways of placing k 2 X 2 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

LINKS

Table of n, a(n) for n=0..83.

FORMULA

T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).

T(n, k) = T(n-1, k) + T(n-2, k-1) - C((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) except when n or k even then T(n, k) = T(n-1, k) + T(n-2, k-1) with T(0, 0) = 1, T(n, 0) = 0 for n<0 and T(n, k) = 0 for k < 0 and k  > floor(n/2). - Johannes W. Meijer, Aug 24 2013

EXAMPLE

The first few rows of triangle T(n, k) are:

n/k: 0, 1, 2, 3

0:   1

1:   1

2:   1, 1

3:   1, 1

4:   1, 2, 1

5:   1, 2, 2

6:   1, 3, 4, 1

7:   1, 3, 6, 2

MAPLE

From Johannes W. Meijer, Aug 24 2013: (Start)

T := proc(n, k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: A034851(n-k, k) end: A034851 := proc(n, k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1, (k-1)/2) else t := 0; fi; A034851(n-1, k-1) + A034851(n-1, k)-t; end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..16);  # End first program

T := proc(n, k) option remember: if n < 0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: if n=0 then return(1) fi: if type(n, even) or type(k, even) then procname(n-1, k) + procname(n-2, k-1) else procname(n-1, k) + procname(n-2, k-1) - binomial((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..16); # End second program (End)

CROSSREFS

Cf. A034851, A005685, A005688, A068930, A102526, A102543, A192928.

Sequence in context: A029320 A187450 A187449 * A243928 A286363 A233573

Adjacent sequences:  A102538 A102539 A102540 * A102542 A102543 A102544

KEYWORD

nonn,tabf

AUTHOR

Gerald McGarvey, Feb 24 2005

EXTENSIONS

Definition edited, incorrect formula deleted, keyword corrected and extended by Johannes W. Meijer, Aug 24 2013

STATUS

approved

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Last modified October 19 15:49 EDT 2018. Contains 316365 sequences. (Running on oeis4.)