A367636 a(n) is the number of distinct combinations that can be created by painting the sections on a shape with n divisions that rotates around its center and consists of 4 identical arms at 90-degree intervals.

2, 12, 140, 2088, 32912, 524832, 8390720, 134226048, 2147516672, 34359869952, 549756339200, 8796095121408, 140737496748032, 2251799847247872, 36028797153198080, 576460752840327168, 9223372039002324992, 147573952598266478592, 2361183241469182607360, 37778931863094601187328

Offset: 1

Author

Kadir E. Celik, Alp Giray Datlar, Iskender Ozturk, and Gulnur Ozbek, Nov 25 2023

Keywords

• nonn
• easy

Comments

A shape/object consists of n divisions (cells) that rotates around its center and consists of 4 identical arms at 90-degree intervals.

Each division (cell) can be unpainted (white) or painted (black).

(4n-3) is the number of divisions (cells) on the object/shape which consists of 4 identical arms at 90-degree intervals.

Examples

			In the figures below, "[ ]" represents an unpainted cell; "[o]" represents a painted cell.
For n = 1, there are a(1) = 2 combinations:
.
  [ ]  [o]
.
For n = 2, there are a(2) = 12 combinations:
.
    [ ]         [ ]         [ ]        [ ]
 [ ][ ][ ]   [ ][ ][o]   [ ][o][ ]  [ ][o][o]
    [ ]         [ ]         [ ]        [ ]
.
    [ ]         [o]         [o]        [o]
 [o][ ][o]   [ ][ ][o]   [o][ ][ ]  [ ][o][ ]
    [ ]         [ ]         [o]        [o]
.
    [ ]         [o]         [o]        [o]
 [o][o][ ]   [o][o][ ]   [o][ ][o]  [o][o][o]
    [o]         [o]         [o]        [o]
.
For n = 3, there are a(3) = 140 combinations:
.
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
 [ ][ ][ ][ ][ ]   [ ][ ][ ][o][ ]   [ ][ ][ ][ ][o]  [ ][ ][o][ ][ ]
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
.
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
 [ ][ ][o][o][ ]   [ ][ ][o][ ][o]   [ ][ ][ ][o][o]  [ ][o][ ][o][ ]
       [ ]               [ ]               [ ]              [ ]
       [ ]               [ ]               [ ]              [ ]
...
		

References

• A. Nesin, Matematik ve sonsuz [Math and infinity], Nesin Yayıncılık, 2019, pages 137-143.

Links

• Gulnur Ozbek, Illustrations for initial terms (with painted cells black, unpainted cells white).
• Index entries for linear recurrences with constant coefficients, signature (22,-104,128).

Programs

Mathematica

CoefficientList[Series[2*(1 - 16*x + 42*x^2)/((1 - 2*x)*(1 - 4*x)*(1 - 16*x)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Dec 10 2023 *)

Formula

a(n) = 2^(4n-5) + 2^(2n-3) + 2^(n-1).

a(n) is the sum of the terms in the n-th row of the following triangle, where k is the number of divisions (cells) which are colored/painted black.

.

n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 4n-3

---+--------------------------------------------------------------------------

1 | 1 1

2 | 1 2 3 3 2 1

3 | 1 3 10 22 34 34 22 10 3 1

4 | 1 4 21 73 184 327 434 434 327 184 73 21 4 1

5 | 1 5 36 172 604 1556 3108 4876 6098 6098 4876 3108 1556 604 172 36 5 1

...|

n | 1 n ...

The term at the intersection of any row and column is

C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4

+ C([(4n-3)/4],[k/4])/2 for k == 0 or 1 (mod 4),

C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4 for k == 2 or 3 (mod 4)

where [] is the floor function.

G.f.: 2*x*(1 - 16*x + 42*x^2)/((1 - 2*x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Dec 03 2023