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

Author

Alex Ratushnyak, Jul 08 2012

Keywords

• nonn
• easy

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

• David Radcliffe, Table of n, a(n) for n = 1..10000

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=', ')