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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.

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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
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