OFFSET
0,9
COMMENTS
The row sums of this triangle are A102543. The antidiagonal sums are given by A192928 and the backwards antidiagonal sums are given by A228571.
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.
Also the number of equivalence classes of ways of placing k 3 X 3 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
T(n, k) = A034851(n-2*k, k), n >= 0 and 0 <= k <= floor(n/3).
T(n, k) = T(n-1, k) + T(n-3, k-1) - C((n-4)/2 - 2*(k-1)/2, (k-1)/2) where the last term is present only if n even and k odd; T(0, 0) = 1, T(1, 0) = 1, T(2, 0) = 1, T(n, k) = 0 for n < 0 and T(n, k) = 0 for k < 0 and k > floor(n/3).
EXAMPLE
The first few rows of triangle T(n, k) are:
n/k: 0, 1, 2, 3
0: 1
1: 1
2: 1
3: 1, 1
4: 1, 1
5: 1, 2
6: 1, 2, 1
7: 1, 3, 2
8: 1, 3, 4
9: 1, 4, 6, 1
10: 1, 4, 9, 2
11: 1, 5, 12, 6
MAPLE
T := proc(n, k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/3) then return(0) fi: A034851(n-2*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/3)), n=0..18); # End first program
T := proc(n, k) option remember: if n=0 and k=0 or n=1 and k=0 or n=2 and k=0 then return(1) fi: if k <0 or k > floor(n/3) then return(0) fi: if type(n, even) and type(k, odd) then procname(n-1, k) + procname(n-3, k-1) - binomial((n-4)/2-2*(k-1)/2, (k-1)/2) else procname(n-1, k) + procname(n-3, k-1) fi: end: seq(seq(T(n, k), k=0..floor(n/3)), n=0..18); # End second program
MATHEMATICA
T[n_, k_] := (Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
PROG
(PARI)
T(n, k)={(binomial(n-2*k, k) + (k%2==0||n%2==1)*binomial((n-2*k-n%2)/2, k\2))/2}
for(n=1, 20, for(k=0, (n\3), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 30 2017
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Johannes W. Meijer, Aug 26 2013
STATUS
approved