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A238190 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows. 21
1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 8, 1, 4, 12, 3, 1, 4, 18, 8, 1, 5, 24, 22, 1, 5, 32, 40, 6, 1, 6, 40, 73, 22, 1, 6, 50, 112, 66, 1, 7, 60, 172, 146, 10, 1, 7, 72, 240, 292, 48, 1, 8, 84, 335, 516, 174, 1, 8, 98, 440, 860, 448, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,6

LINKS

Andrew Howroyd, Table of n, a(n) for n = 3..974

Christopher Hunt Gribble, C++ program

EXAMPLE

The first 13 rows of T(n,k) are:

.\ k    0     1     2     3     4     5

n

3       1     1

4       1     1

5       1     2

6       1     2     2

7       1     3     4

8       1     3     8

9       1     4    12     3

10      1     4    18     8

11      1     5    24    22

12      1     5    32    40     6

13      1     6    40    73    22

14      1     6    50   112    66

15      1     7    60   172   146    10

MATHEMATICA

T[n_, k_] := (2^k Binomial[n - 2k, k] + (Boole[EvenQ[k]] + Boole[OddQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k + 1, 2] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-Fran├žois Alcover, Oct 06 2017, after Andrew Howroyd *)

PROG

(C++) See Gribble link.

(PARI)

T(n, k)={(2^k*binomial(n-2*k, k) + ((k%2==0)+(n%2==1||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-2*k-(n%2))/2, k\2))/4}

for(n=2, 20, for(k=0, floor(n/3), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 29 2017

CROSSREFS

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Sequence in context: A201593 A029287 A055184 * A227925 A035388 A255716

Adjacent sequences:  A238187 A238188 A238189 * A238191 A238192 A238193

KEYWORD

tabf,nonn

AUTHOR

Christopher Hunt Gribble, Feb 19 2014

EXTENSIONS

Link to C++ program and xrefs updated by Christopher Hunt Gribble, Apr 25 2015

Terms a(51) and beyond from Andrew Howroyd, May 29 2017

STATUS

approved

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Last modified February 19 19:30 EST 2020. Contains 332047 sequences. (Running on oeis4.)