This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A367636 #53 Jan 20 2024 09:17:04
%S A367636 2,12,140,2088,32912,524832,8390720,134226048,2147516672,34359869952,
%T A367636 549756339200,8796095121408,140737496748032,2251799847247872,
%U A367636 36028797153198080,576460752840327168,9223372039002324992,147573952598266478592,2361183241469182607360,37778931863094601187328
%N 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.
%C A367636 A shape/object consists of n divisions (cells) that rotates around its center and consists of 4 identical arms at 90-degree intervals.
%C A367636 Each division (cell) can be unpainted (white) or painted (black).
%C A367636 (4n-3) is the number of divisions (cells) on the object/shape which consists of 4 identical arms at 90-degree intervals.
%D A367636 A. Nesin, Matematik ve sonsuz [Math and infinity], Nesin Yayıncılık, 2019, pages 137-143.
%H A367636 Gulnur Ozbek, <a href="/A367636/a367636_1.pdf">Illustrations for initial terms</a> (with painted cells black, unpainted cells white).
%H A367636 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (22,-104,128).
%F A367636 a(n) = 2^(4n-5) + 2^(2n-3) + 2^(n-1).
%F A367636 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.
%F A367636 .
%F A367636 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 4n-3
%F A367636 ---+--------------------------------------------------------------------------
%F A367636 1 | 1 1
%F A367636 2 | 1 2 3 3 2 1
%F A367636 3 | 1 3 10 22 34 34 22 10 3 1
%F A367636 4 | 1 4 21 73 184 327 434 434 327 184 73 21 4 1
%F A367636 5 | 1 5 36 172 604 1556 3108 4876 6098 6098 4876 3108 1556 604 172 36 5 1
%F A367636 ...|
%F A367636 n | 1 n ...
%F A367636 The term at the intersection of any row and column is
%F A367636 C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4
%F A367636 + C([(4n-3)/4],[k/4])/2 for k == 0 or 1 (mod 4),
%F A367636 C((4n-3),k)/4 + C([(4n-3)/2],[k/2])/4 for k == 2 or 3 (mod 4)
%F A367636 where [] is the floor function.
%F A367636 G.f.: 2*x*(1 - 16*x + 42*x^2)/((1 - 2*x)*(1 - 4*x)*(1 - 16*x)). - _Stefano Spezia_, Dec 03 2023
%e A367636 In the figures below, "[ ]" represents an unpainted cell; "[o]" represents a painted cell.
%e A367636 For n = 1, there are a(1) = 2 combinations:
%e A367636 .
%e A367636 [ ] [o]
%e A367636 .
%e A367636 For n = 2, there are a(2) = 12 combinations:
%e A367636 .
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ][ ][ ] [ ][ ][o] [ ][o][ ] [ ][o][o]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 .
%e A367636 [ ] [o] [o] [o]
%e A367636 [o][ ][o] [ ][ ][o] [o][ ][ ] [ ][o][ ]
%e A367636 [ ] [ ] [o] [o]
%e A367636 .
%e A367636 [ ] [o] [o] [o]
%e A367636 [o][o][ ] [o][o][ ] [o][ ][o] [o][o][o]
%e A367636 [o] [o] [o] [o]
%e A367636 .
%e A367636 For n = 3, there are a(3) = 140 combinations:
%e A367636 .
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ][ ][ ][ ][ ] [ ][ ][ ][o][ ] [ ][ ][ ][ ][o] [ ][ ][o][ ][ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 .
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ][ ][o][o][ ] [ ][ ][o][ ][o] [ ][ ][ ][o][o] [ ][o][ ][o][ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 [ ] [ ] [ ] [ ]
%e A367636 ...
%t A367636 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 *)
%K A367636 nonn,easy
%O A367636 1,1
%A A367636 _Kadir E. Celik_, _Alp Giray Datlar_, _Iskender Ozturk_, and _Gulnur Ozbek_, Nov 25 2023