A217730 -id:A217730 - cp's OEIS frontend

A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 0, 5, 10, 9, 0, 0, 0, 5, 15, 19, 9, 0, 0, 0, 0, 20, 34, 28, 0, 0, 0, 0, 0, 20, 54, 62, 28, 0, 0, 0, 0, 0, 0, 74, 116, 90, 0, 0, 0, 0, 0, 0, 0, 74, 190, 206, 90, 0, 0, 0, 0, 0, 0, 0, 0, 264, 396, 296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 264, 660, 692, 296, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Arithmetic hexagon of E. Lucas.

			Square array begins:
  1, 1, 1,  1,  1,   0,   0,   0,   0,   0, 0, ... row n=0
  1, 2, 3,  4,  5,   5,   0,   0,   0,   0, 0, ... row n=1
  1, 3, 6, 10, 15,  20,  20,   0,   0,   0, 0, ... row n=2
  0, 3, 9, 19, 34,  54,  74,  74,   0,   0, 0, ... row n=3
  0, 0, 9, 28, 62, 116, 190, 264, 264,   0, 0, ... row n=4
  0, 0, 0, 28, 90, 206, 396, 660, 924, 924, 0, ... row n=5
  ...
Array, read by rows, with 0 omitted:
   1,   1,   1,   1,    1
   1,   2,   3,   4,    5,    5
   1,   3,   6,  10,   15,   20,   20
        3,   9,  19,   34,   54,   74,   74
             9,  28,   62,  116,  190,  264,  264
                 28,   90,  206,  396,  660,  924,  924
                       90,  296,  692, 1352, 2276, 3200, 3200
  ...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris, 1958, Tome 1, p. 89.

E. Lucas, Théorie des nombres, Gauthier-Villars, Paris 1891, Tome 1, p. 89.

Cf. A007052, A094803, A094806, A094817, A094821

Cf. similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230.

T(n,n) = A094817(n), for n > 0.
T(n+1,n) = T(n+2,n) = A094803(n).
T(n,n+1) = A007052(n).
T(n,n+2) = A094821(n+1).
T(n,n+3) = T(n,n+4) = A094806(n).
Sum_{k=0..n} T(n-k,k) = A217730(n). - Philippe Deléham, Mar 22 2013

A332204 a(n) is the real part of f(n) defined by f(0) = 0, and f(n+1) = f(n) + g((1+i)^(A065359(n) mod 8)) (where g(z) = z/gcd(Re(z), Im(z)) and i denotes the imaginary unit).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 17, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 31, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 49, 49, 50

Offset: 0

Rémy Sigrist, Feb 07 2020

The representation of {f(n)} resembles a Koch curve (see illustrations in Links section).
The sequence A065359 mod 8 gives the direction at each step as follows:
         3      2      1
            \   |   /
              \ | /
                \|/
         4 ------.------ 0
               /|\
             /  |  \
           /    |    \
         5       6       7
We can also build {f(n)} with A096268 as follows:
- start at the origin looking to the right,
- for k=0, 1, ...:
    - move forward to the next lattice point
      (this point is at distance 1 or sqrt(2)),
    - if A096268(k)=0
      then turn 45 degrees to the left
      otherwise turn 90 degrees to the right,
- this connects the first differences of A065359 and A096268.

			The first terms, alongside f(n) and A065359(n), are:
  n   a(n)  f(n)   A065359(n)
  --  ----  -----  ----------
   0     0      0           0
   1     1      1           1
   2     2    2+i          -1
   3     3      3           0
   4     4      4           1
   5     5    5+i           2
   6     5  5+2*i           0
   7     6  6+2*i           1
   8     7  7+3*i          -1
   9     8  8+2*i           0
  10     9  9+2*i          -2
  11     9    9+i          -1
  12    10     10           0
  13    11     11           1
  14    12   12+i          -1
  15    13     13           0
  16    14     14           1

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Larry Riddle, Koch Curve
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, Representation of f(n) in the complex plan for n = 0..2^14
Rémy Sigrist, PARI program for A332204
Index entries for sequences related to coordinates of 2D curves

Cf. A065359, A096268, A217730, A332205 (imaginary part), A332206 (where f is real).

A065359[0] = 0;
A065359[n_] := -Total[(-1)^PositionIndex[Reverse[IntegerDigits[n, 2]]][1]];
g[z_] := z/GCD[Re[z], Im[z]];
Module[{n = 0}, Re[NestList[# + g[(1+I)^A065359[n++]] &, 0, 100]]] (* Paolo Xausa, Aug 28 2024 *)

PARI
```
\\ See Links section.
```

a(2^k) = A217730(k) for any k >= 0.

a(4^k+m) + a(m) = A217730(2*k) for any k >= 0 and m = 0..4^k.

cp's OEIS Frontend

A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula