cp's OEIS Frontend

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.

Showing 1-6 of 6 results.

A217010 Permutation of natural numbers arising from applying the walk of a square spiral (e.g., A214526) to the data of right triangular type-1 spiral (defined in A214230).

Original entry on oeis.org

1, 2, 13, 3, 5, 6, 7, 8, 9, 10, 12, 32, 61, 33, 14, 4, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 11, 31, 60, 98, 145, 99, 62, 34, 15, 17, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 28, 30, 59, 97, 144, 200, 265, 201, 146, 100, 63, 35, 16, 39, 71
Offset: 1

Views

Author

Alex Ratushnyak, Sep 23 2012

Keywords

Examples

			Triangular spiral (A214230) begins:
.
  56
   | \
  55  57
   |     \
  54  29  58
   |   | \   \
  53  28  30  59
   |   |     \   \
  52  27  11  31  60
   |   |   | \   \   \
  51  26  10  12  32  61
   |   |   |     \   \   \
  50  25   9   2  13  33  62
   |   |   |   | \   \   \   \
  49  24   8   1   3  14  34  63
   |   |   |         \   \   \   \
  48  23   7---6---5---4  15  35  64
   |   |                     \   \   \
  47  22--21--20--19--18--17--16  36  65
   |                                 \   \
  46--45--44--43--42--41--40--39--38--37  66
                                             \
  78--77--76--75--74--73--72--71--70--69--68--67
.
Square spiral (defining order in which elements are fetched) begins:
.
  49  26--27--28--29--30--31
   |   |                   |
  48  25  10--11--12--13  32
   |   |   |           |   |
  47  24   9   2---3  14  33
   |   |   |   |   |   |   |
  46  23   8   1   4  15  34
   |   |   |       |   |   |
  45  22   7---6---5  16  35
   |   |               |   |
  44  21--20--19--18--17  36
   |                       |
  43--42--41--40--39--38--37

Crossrefs

Cf. A090861, A214526, A214230.

Programs

  • Python
    SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(0, -1,  1,  1)    # up
    walk( 1, 1, -1,  0)    # right-down
    if posX==SIZE-1:
        break
    walk(-1, 0,  0, -1)    # left
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217291 Permutation of natural numbers arising from applying the walk of right triangular type-1 spiral (defined in A214230) to the data of square spiral (e.g. A214526).

Original entry on oeis.org

1, 2, 4, 16, 5, 6, 7, 8, 9, 10, 27, 11, 3, 15, 35, 63, 36, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 51, 84, 52, 28, 12, 14, 34, 62, 98, 142, 99, 64, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 83, 124, 173, 125, 85, 53, 29, 13, 33, 61, 97, 141, 193, 253
Offset: 1

Views

Author

Alex Ratushnyak, Sep 30 2012

Keywords

Crossrefs

Cf. A214230, A214526, A217010.

Programs

  • Python
    SIZE = 29    # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX:
    walk(0, -1, 1, 0)    # up
    walk(1, 0, 0, 1)     # right
    walk(0, 1, -1, 0)    # down
    walk(-1, 0, 0, -1)   # left
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk2(0, -1,  1,  1)    # up
    walk2( 1, 1, -1,  0)    # right-down
    if posX==SIZE-1:
        break
    walk2(-1, 0,  0, -1)    # left

A215468 Sum of the 8 nearest neighbors of n in a rotated-square spiral with positive integers.

Original entry on oeis.org

50, 62, 72, 86, 76, 84, 122, 88, 144, 104, 166, 120, 152, 160, 144, 218, 160, 168, 248, 184, 192, 278, 208, 216, 260, 268, 240, 248, 346, 264, 272, 280, 384, 296, 304, 312, 422, 328, 336, 344, 400, 408, 368, 376, 384, 506, 400, 408, 416, 424, 552, 440, 448, 456, 464, 598
Offset: 1

Views

Author

Alex Ratushnyak, Aug 11 2012

Keywords

Examples

			Spiral begins:
                     85
                     /
                    /
                  84 61-62
                  /  /    \
                 /  /      \
               83 60 41-42 63
               /  /  /    \  \
              /  /  /      \  \
            82 59 40 25-26 43 64
            /  /  /  /    \  \  \
           /  /  /  /      \  \  \
         81 58 39 24 13-14 27 44 65
         /  /  /  /  /    \  \  \  \
        /  /  /  /  /      \  \  \  \
      80 57 38 23 12  5--6 15 28 45 66
      /  /  /  /  /  /    \  \  \  \  \
     /  /  /  /  /  /      \  \  \  \  \
   79 56 37 22 11  4  1--2  7 16 29 46 67
     \  \  \  \  \  \   /  /  /  /  /  /
      \  \  \  \  \  \ /  /  /  /  /  /
      78 55 36 21 10  3  8 17 30 47 68
        \  \  \  \  \   /  /  /  /  /
         \  \  \  \  \ /  /  /  /  /
         77 54 35 20  9 18 31 48 69
           \  \  \  \   /  /  /  /
            \  \  \  \ /  /  /  /
            76 53 34 19 32 49 70
              \  \  \   /  /  /
               \  \  \ /  /  /
               75 52 33 50 71
                 \  \   /  /
                  \  \ /  /
                  74 51 72
                    \   /
                     \ /
                     73
.
The 8 nearest neighbors of 4 are 1,3,5,10,11,12,21,23, their sum is 86, so a(4)=86.

Links

Crossrefs

Coordinates (but 0-based): A010751, A305258.
Other spiral sums: A214176, A214177, A214226, A214227, A214230, A214231, A214250, A214251, A214252.

Programs

  • Python
    SIZE=17  # must be odd
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
saveX = [0]* (SIZE*SIZE+1)
saveY = [0]* (SIZE*SIZE+1)
grid[posY*SIZE+posX]=1
saveX[1]=posX
saveY[1]=posY
posX += 1
grid[posY*SIZE+posX]=2
saveX[2]=posX
saveY[2]=posY
n = 3
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX!=SIZE-1:
    walk(-1,  1, -1, -1)    # down-left
    walk(-1, -1,  1, -1)    # up-left
    walk( 1, -1,  1,  0)    # up-right
    walk( 1,  0,  1,  1)    # right
    walk( 1,  1, -1,  1)    # down-right
for s in range(1, n):
    posX = saveX[s]
    posY = saveY[s]
    i,j = grid[(posY-1)*SIZE+posX-1], grid[(posY-1)*SIZE+posX+1]
    u,v = grid[(posY+1)*SIZE+posX-1], grid[(posY+1)*SIZE+posX+1]
    if i==0 or j==0 or u==0 or v==0:
        break
    k = grid[(posY-1)*SIZE+posX  ] + grid[(posY+1)*SIZE+posX  ]
    k+= grid[ posY   *SIZE+posX-1] + grid[ posY   *SIZE+posX+1]
    print(i+j+u+v+k, end=' ')
print()
for y in range(SIZE):
    for x in range(SIZE):
        print('%3d' % grid[y*SIZE+x], end=' ')
    print()
  • Python
    def spiral(x, y):
    r = abs(x) + abs(y)
    return 1 + 2*r*r + (y-r if x > 0 else r-y)
def A215468(n):
    x = A010751(n-1)
    y = A305258(n-1)
    return sum(spiral(x+i, y+j) for i in (-1, 0, 1) for j in (-1, 0, 1)
               if (i, j) != (0, 0)) # David Radcliffe, Aug 05 2025

A214251 Sum of the eight nearest neighbors of n in a right triangular type-2 spiral with positive integers.

Original entry on oeis.org

62, 64, 69, 125, 94, 111, 170, 118, 105, 116, 169, 132, 131, 151, 192, 284, 217, 201, 206, 220, 258, 353, 265, 234, 227, 235, 243, 269, 349, 285, 275, 283, 291, 299, 328, 387, 515, 412, 378, 374, 382, 390, 398, 421, 477, 608, 484, 435, 419, 427, 435
Offset: 1

Views

Author

Alex Ratushnyak, Jul 08 2012

Keywords

Comments

Right triangular type-1 spiral (A214230): implements the sequence Up, Right-down, Left.
Right triangular type-2 spiral: Left, Up, Right-down.
Right triangular type-3 spiral (A214252): Right-down, Left, Up.

Examples

			Right triangular type-2 spiral begins:
67
66  68
65  37  69
64  36  38  70
63  35  16  39  71
62  34  15  17  40  72
61  33  14   4  18  41  73
60  32  13   3   5  19  42  74
59  31  12   2   1   6  20  43  75
58  30  11  10   9   8   7  21  44  76
57  29  28  27  26  25  24  23  22  45  77
56  55  54  53  52  51  50  49  48  47  46  78
The eight nearest neighbors of 5 are 1, 2, 3, 4, 18, 41, 19, 6. Their sum is a(5)=94.

Crossrefs

Cf. A214230.
Cf. A214252.

Programs

  • Python
    SIZE=29  # must be odd
grid = [0] * (SIZE*SIZE)
saveX = [0]* (SIZE*SIZE)
saveY = [0]* (SIZE*SIZE)
saveX[1] = saveY[1] = posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(-1, 0,  0, -1)    # left
    walk(0, -1,  1,  1)    # up
    if posY==0:
        break
    walk( 1, 1, -1,  0)    # right-down
for n in range(1, 92):
    posX = saveX[n]
    posY = saveY[n]
    k = grid[(posY-1)*SIZE+posX] + grid[(posY+1)*SIZE+posX]
    k+= grid[(posY-1)*SIZE+posX-1] + grid[(posY-1)*SIZE+posX+1]
    k+= grid[(posY+1)*SIZE+posX-1] + grid[(posY+1)*SIZE+posX+1]
    k+= grid[posY*SIZE+posX-1] + grid[posY*SIZE+posX+1]
    print(k, end=', ')

A214252 Sum of the eight nearest neighbors of n in a right triangular type-3 spiral with positive integers.

Original entry on oeis.org

62, 88, 63, 89, 76, 102, 170, 127, 126, 152, 223, 159, 140, 139, 159, 221, 175, 171, 179, 202, 249, 353, 274, 252, 254, 262, 279, 323, 430, 330, 293, 283, 291, 299, 307, 336, 425, 352, 339, 347, 355, 363, 371, 403, 468, 608, 493, 453, 446, 454, 462, 470, 478, 504
Offset: 1

Views

Author

Alex Ratushnyak, Jul 08 2012

Keywords

Comments

Right triangular type-1 spiral (A214230): implements the sequence Up, Right-down, Left.
Right triangular type-2 spiral (A214251): Left, Up, Right-down.
Right triangular type-3 spiral: Right-down, Left, Up.

Examples

			Right triangular type-3 spiral begins:
78
77  46
76  45  47
75  44  22  48
74  43  21  23  49
73  42  20   7  24  50
72  41  19   6   8  25  51
71  40  18   5   1   9  26  52
70  39  17   4   3   2  10  27  53
69  38  16  15  14  13  12  11  28  54
68  37  36  35  34  33  32  31  30  29  55
67  66  65  64  63  62  61  60  59  58  57  56
The eight nearest neighbors of 5 are 1, 3, 4, 17, 18, 19, 6, 8. Their sum is a(5)=76.

Crossrefs

Cf. A214230.
Cf. A214251.

Programs

  • Python
    SIZE=28  # must be even
grid = [0] * (SIZE*SIZE)
saveX = [0]* (SIZE*SIZE)
saveY = [0]* (SIZE*SIZE)
saveX[1] = saveY[1] = posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posY!=0:
    walk( 1, 1, -1,  0)    # right-down
    walk(-1, 0,  0, -1)    # left
    walk(0, -1,  1,  1)    # up
for n in range(1, 92):
    posX = saveX[n]
    posY = saveY[n]
    k = grid[(posY-1)*SIZE+posX] + grid[(posY+1)*SIZE+posX]
    k+= grid[(posY-1)*SIZE+posX-1] + grid[(posY-1)*SIZE+posX+1]
    k+= grid[(posY+1)*SIZE+posX-1] + grid[(posY+1)*SIZE+posX+1]
    k+= grid[posY*SIZE+posX-1] + grid[posY*SIZE+posX+1]
    print(k, end=' ')

A214231 Sum of the four nearest neighbors of n in a right triangular type-1 spiral with positive integers.

Original entry on oeis.org

19, 35, 33, 52, 32, 33, 58, 41, 45, 58, 98, 75, 70, 74, 90, 127, 89, 81, 85, 89, 93, 136, 101, 105, 109, 113, 117, 139, 197, 156, 142, 146, 150, 154, 158, 183, 238, 182, 165, 169, 173, 177, 181, 185, 189, 250, 197, 201, 205, 209, 213, 217, 221, 225, 256, 332
Offset: 1

Views

Author

Alex Ratushnyak, Jul 08 2012

Keywords

Comments

Nearby numbers on diagonals are not counted as neighbors for this sequence.
Right triangular type-1 spiral implements the sequence Up, Right-down, Left.

Examples

			Right triangular spiral begins:
56
55  57
54  29  58
53  28  30  59
52  27  11  31  60
51  26  10  12  32  61
50  25   9   2  13  33  62
49  24   8   1   3  14  34  63
48  23   7   6   5   4  15  35  64
47  22  21  20  19  18  17  16  36  65
46  45  44  43  42  41  40  39  38  37  66
78  77  76  75  74  73  72  71  70  69  68  67
The four nearest neighbors of 3 are 1, 13, 14, 5. Their sum is a(3)=33.

Crossrefs

Cf. A214230 - sum of the 8 nearest neighbors, Python program.
Cf. A214227.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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