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A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition). 7
0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, only one cell is turned ON, so a(1) = 1.
If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).
Has the same rules as A256534 but here a(1) = 1 not 4.
Has a smoother behavior than A160414 with which shares infinitely many terms (see example).
A256531, the first differences, gives the number of cells turned ON at n-th stage.
LINKS
FORMULA
For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i
EXAMPLE
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21, 49;
61, 97, 157, 225;
237, 273, 333, 417, 525, 657, 813, 961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ _ _ _ _ |_| |
. |_ _| | _ _ _ _ | | _ _ _ _ | |_ _|
. | | | _ _ | | | | _ _ | | |
. | | | | _|_|_|_|_|_|_ | | | |
. | | | |_| _ _ _ _ |_| | | |
. | | |_ _| | _|_|_ | |_ _| | |
. | |_ _ _| |_| _ |_| |_ _ _| |
. | _ _ _| _| |_| |_ |_ _ _ |
. | | _ _| | |_ _ _| | |_ _ | |
. | | | _| |_ _| |_ _| |_ | | |
. | | | | |_ _ _ _ _ _ _| | | | |
. | | | |_ _| | | | | |_ _| | | |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
MATHEMATICA
With[{z=7}, Join[{0}, Flatten[Array[(2^#-1)^2+12Range[0, 2^(#-1)-1]^2&, z]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)
PROG
(GW-BASIC) 10' a256530 First 2^z-1 terms: 20 z=6: defdbl a: for i=1 to z: for j=0 to 2^(i-1)-1: n=n+1: a(n)=(2^i-1)^2 + 3*(2*j)^2: print a(n); : next j: next i: end
CROSSREFS
Sequence in context: A241747 A133762 A160414 * A363634 A118130 A144482
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Apr 21 2015
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)