OFFSET
0,2
COMMENTS
Draw a square spiral on a piece of graph paper, and label the cells starting at the center with the positive (resp. nonnegative) numbers. This produces two versions of the labeled square spiral, shown in the Example section below.
The spiral may proceed clockwise or counterclockwise, and the first arm of the spiral may be along any of the four axes, so there are eight versions of each spiral. However, this has no effect on the resulting sequences, and it is enough to consider just two versions of the square spiral (starting at 1 or starting at 0).
The present sequence is obtained by reading alternate entries on the X-axis (say) of the square spiral started at 1.
The cross-references section lists many sequences that can be read directly off the two spirals. Many other sequences can be obtained from them by using them to extract subsequences from other important sequences. For example, the subsequence of primes indexed by the present sequence gives A317187.
a(n) is also the number of free polyominoes with n + 4 cells whose difference between length and width is n. In this comment the length is the longer of the two dimensions and the width is the shorter of the two dimensions (see the examples of polyominoes). Hence this is also the diagonal 4 of A379625. - Omar E. Pol, Jan 24 2025
From John Mason, Feb 19 2025: (Start)
The sequence enumerates polyominoes of width 2 having precisely 2 horizontal bars. By classifying such polyominoes according to the following templates, it is possible to define a formula that reduces to the one below:
.
OO O O
O OO OO
O O O
O O OO
OO OO O
.
(End)
LINKS
FORMULA
From Daniel Forgues, Aug 01 2018: (Start)
a(n) = (1/4) * (4 * n^2 + 2 * n + (-1)^n + 3), n >= 0.
a(0) = 1; a(n) = - a(n-1) + 2 * n^2 - n + 2, n >= 1.
a(0) = 1; a(1) = 2; a(2) = 6; a(3) = 11; a(n) = 2 * a(n-1) - 2 * a(n-3) + a(n-4), n >= 4.
G.f.: (- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)). (End)
E.g.f.: ((2 + 3*x + 2*x^2)*cosh(x) + (1 + 3*x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Apr 24 2024
EXAMPLE
The square spiral when started with 1 begins:
.
100--99--98--97--96--95--94--93--92--91
|
65--64--63--62--61--60--59--58--57 90
| | |
66 37--36--35--34--33--32--31 56 89
| | | | |
67 38 17--16--15--14--13 30 55 88
| | | | | | |
68 39 18 5---4---3 12 29 54 87
| | | | | | | | |
69 40 19 6 1---2 11 28 53 86
| | | | | | | |
70 41 20 7---8---9--10 27 52 85
| | | | | |
71 42 21--22--23--24--25--26 51 84
| | | |
72 43--44--45--46--47--48--49--50 83
| |
73--74--75--76--77--78--79--80--81--82
.
For the square spiral when started with 0, subtract 1 from each entry. In the following diagram this spiral has been reflected and rotated, but of course that makes no difference to the sequences:
.
99 64--65--66--67--68--69--70--71--72
| | |
98 63 36--37--38--39--40--41--42 73
| | | | |
97 62 35 16--17--18--19--20 43 74
| | | | | | |
96 61 34 15 4---5---6 21 44 75
| | | | | | | | |
95 60 33 14 3 0 7 22 45 76
| | | | | | | | | |
94 59 32 13 2---1 8 23 46 77
| | | | | | | |
93 58 31 12--11--10---9 24 47 78
| | | | | |
92 57 30--29--28--27--26--25 48 79
| | | |
91 56--55--54--53--52--51--50--49 80
| |
90--89--88--87--86--85--84--83--82--81
.
From Omar E. Pol, Jan 24 2025: (Start)
For n = 0 there is only one free polyomino with 0 + 4 = 4 cells whose difference between length and width is 0 as shown below, so a(0) = 1.
_ _
|_|_|
|_|_|
.
For n = 1 there are two free polyominoes with 1 + 4 = 5 cells whose difference between length and width is 1 as shown below, so a(1) = 2.
_ _ _ _
|_|_| |_|_|
|_|_| |_|_
|_| |_|_|
.
(End)
MATHEMATICA
a[n_] := n^2 + n - Floor[(n - 1)/2]; Array[a, 50, 0] (* Robert G. Wilson v, Aug 01 2018 *)
LinearRecurrence[{2, 0, -2 , 1}, {1, 2, 6, 11}, 50] (* or *)
CoefficientList[Series[(- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)), {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)
CROSSREFS
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. also A317187.
KEYWORD
nonn,easy,changed
AUTHOR
N. J. A. Sloane, Jul 27 2018
STATUS
approved