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.

A217014 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of triangular horizontal-last spiral (defined in A214226).

Original entry on oeis.org

1, 7, 22, 8, 2, 3, 4, 6, 20, 42, 21, 44, 75, 45, 23, 9, 11, 12, 13, 14, 15, 5, 19, 41, 71, 109, 72, 43, 74, 113, 160, 114, 76, 46, 24, 10, 28, 29, 30, 31, 32, 33, 34, 16, 18, 40, 70, 108, 154, 208, 155, 110, 73, 112, 159, 214
Offset: 1

Views

Author

Alex Ratushnyak, Sep 23 2012

Keywords

Crossrefs

Cf. A090861, A214526, A214226, A217010.

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(1,  1, -1,  0)    # down-right
    walk(-1, 0,  1, -1)    # left
    walk(-1, 0,  1, -1)    # left
    if posX<2:
        break
    walk(1, -1,  1,  1)    # up-right
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, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217295 Permutation of natural numbers arising from applying the walk of triangular horizontal-last spiral (defined in A214226) to the data of square spiral (e.g. A214526).

Original entry on oeis.org

1, 5, 6, 7, 22, 8, 2, 4, 16, 36, 17, 18, 19, 20, 21, 44, 75, 45, 23, 9, 11, 3, 15, 35, 63, 99, 64, 37, 38, 39, 40, 41, 42, 43, 74, 113, 160, 114, 76, 46, 24, 10, 28, 12, 14, 34, 62, 98, 142, 194, 143, 100, 65, 66, 67, 68, 69, 70, 71, 72, 73, 112, 159, 214, 277
Offset: 1

Views

Author

Alex Ratushnyak, Sep 30 2012

Keywords

Crossrefs

Cf. A214226, A214526, A217014.

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(1,  1, -1,  0)    # down-right
    walk2(-1, 0,  1, -1)    # left
    walk2(-1, 0,  1, -1)    # left
    if posX<2:
        break
    walk2(1, -1,  1,  1)    # up-right

A214230 Sum of the eight nearest neighbors of n in a right triangular type-1 spiral with positive integers.

Original entry on oeis.org

53, 88, 78, 125, 85, 84, 125, 97, 108, 143, 223, 168, 158, 169, 201, 284, 208, 183, 179, 187, 210, 281, 226, 219, 227, 235, 261, 314, 430, 339, 311, 310, 318, 326, 346, 396, 515, 403, 360, 347, 355, 363, 371, 379, 411, 509, 427, 411, 419, 427, 435, 443, 451, 486, 557
Offset: 1

Views

Author

Alex Ratushnyak, Jul 08 2012

Keywords

Comments

Right triangular type-1 spiral implements the sequence Up, Right-down, Left.
Right triangular type-2 spiral (A214251): Left, Up, Right-down.
Right triangular type-3 spiral (A214252): Right-down, Left, Up.
A140064 -- rightwards from 1: 3,14,34...
A064225 -- leftwards from 1: 8,24,49...
A117625 -- upwards from 1: 2,12,31...
A006137 -- downwards from 1: 6,20,43...
A038764 -- left-down from 1: 7,22,46...
A081267 -- left-up from 1: 9,26,52...
A081589 -- right-up from 1: 13, 61, 145...
9*x^2/2 - 19*x/2 + 6 -- right-down from 1: 5,18,40...

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 eight nearest neighbors of 3 are 1, 2, 13, 33, 14, 4, 5, 6. Their sum is a(3)=78.

Links

Crossrefs

Cf. A214251, A214252.
Cf. A214226, A214231.

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(0, -1,  1,  1)    # up
    if posY==0:
        break
    walk( 1, 1, -1,  0)    # right-down
    walk(-1, 0,  0, -1)    # left
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=', ')

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

A214250 Sum of the eight nearest neighbors of n in a triangular "horizontal-first" spiral with positive integers.

Original entry on oeis.org

56, 80, 108, 84, 132, 92, 84, 96, 128, 200, 152, 144, 236, 160, 172, 204, 284, 212, 188, 184, 192, 200, 208, 232, 280, 384, 304, 280, 280, 288, 428, 304, 312, 320, 340, 388, 500, 396, 356, 344, 352, 360, 368, 376, 384, 392, 400, 432, 496, 632, 520, 480, 472, 480
Offset: 1

Views

Author

Alex Ratushnyak, Jul 08 2012

Keywords

Examples

			Spiral begins:
__ __ __ __ __ __ __ 57
__ __ __ __ __ __ 56 31 58
__ __ __ __ __ 55 30 13 32 59
__ __ __ __ 54 29 12  3 14 33 60
__ __ __ 53 28 11  2  1  4 15 34 61
__ __ 52 27 10  9  8  7  6  5 16 35 62
__ 51 26 25 24 23 22 21 20 19 18 17 36 63
50 49 48 47 46 45 44 43 42 41 40 39 38 37 64
The eight nearest neighbors of 4 are: 1, 3, 14, 33, 15, 5, 6, 7. Their sum is a(4)=84.

Crossrefs

Cf. A214226 - same sum in a triangular "horizontal-last" spiral.

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,chkX2,chkY2):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posX==0 or grid[(posY+chkY)*SIZE+posX+chkX]+grid[(posY+chkY2)*SIZE+posX+chkX2]==0:
        return
while 1:
    walk(-1, 0,  1, -1, 0, -1)   # left
    if posX==0:
        break
    walk( 1,-1,  1, 1,  1, 1)    # right+up
    walk( 1, 1, -1,  0, -1, 0)   # right+down
for n in range(1,122):
    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=' ')

A214227 Sum of the four nearest neighbors of n in a triangular spiral with positive integers.

Original entry on oeis.org

24, 32, 20, 28, 56, 44, 64, 48, 60, 88, 60, 56, 60, 64, 68, 84, 128, 100, 92, 96, 136, 104, 108, 112, 132, 176, 132, 120, 124, 128, 132, 136, 140, 144, 148, 172, 232, 188, 172, 176, 180, 184, 240, 192, 196, 200, 204, 208, 236, 296, 236, 216, 220, 224, 228, 232, 236
Offset: 1

Views

Author

Alex Ratushnyak, Jul 07 2012

Keywords

Comments

Nearby numbers on diagonals are not counted as neighbors for this sequence.
The central 1 is at ?

Examples

			Triangular spiral begins:
.
                       43
                       / \
                      /   \
                    42 21 44
                    /  / \  \
                   /  /   \  \
                 41 20  7 22 45
                 /  /  / \  \  \
                /  /  /   \  \  \
              40 19  6  1  8 23 46
              /  /  /    \  \  \  \
             /  /  /      \  \  \  \
           39 18  5--4--3--2  9 24 47
           /  /                \  \  \
          /  /                  \  \  \
        38 17-16-15-14-13-12-11-10 25 48
        /                            \  \
       /                              \  \
     37-36-35-34-33-32-31-30-29-28-27-26 49
                                           \
                                            \
  64-63-62-61-60-59-58-57-56-55-54-53-52-51-50
.
The four nearest neighbors of 3 are 1, 4, 2, 13; their sum is a(3)=20.

Crossrefs

Cf. A214226 (sum of the 8 nearest neighbors).
Showing 1-6 of 6 results.

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