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

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

1,1

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...

Links

Table of n, a(n) for n=1..55.

Example

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.

Prog

(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,

Crossrefs

Cf. A214251, A214252.

Cf. A214226, A214231.

Sequence in context: A144939 A118149 A251144 * A272367 A119289 A124282

Adjacent sequences: A214227 A214228 A214229 * A214231 A214232 A214233

Keyword

nonn,easy

Author

Alex Ratushnyak, Jul 08 2012

Status

approved