A232440 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5), read by rows.

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 4, 1, 5, 6, 1, 5, 9, 1, 6, 12, 1, 1, 6, 16, 2, 1, 7, 20, 6, 1, 7, 25, 10, 1, 8, 30, 19, 1, 8, 36, 28, 1, 1, 9, 42, 44, 3, 1, 9, 49, 60, 9, 1, 10, 56, 85, 19, 1, 10, 64, 110, 38, 1, 11, 72, 146, 66, 1

Offset: 5

Author

Christopher Hunt Gribble, Feb 23 2014

Keywords

• tabf
• nonn

Examples

			The first 9 rows of T(n,k) are:
.\ k    0      1      2
n
5       1      1
6       1      1
7       1      2
8       1      2
9       1      3
10      1      3      1
11      1      4      2
12      1      4      4
13      1      5      6
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 5..989
• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Programs

Mathematica

T[n_, k_] := (Binomial[n - 4k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 4k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 5, 20}, {k, 0, Quotient[n, 5]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

PARI

T(n,k)={(binomial(n-4*k,k) + (k%2==0||n%2==1)*binomial((n-4*k-n%2)/2,k\2))/2}
for(n=5,20,for(k=0,(n\5), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms extended and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

Terms a(27) and beyond from Andrew Howroyd, May 29 2017